Contents · In: Yu. Kabanov, R. Liptser, and J. Stoyanov (editors), From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, pp. 433-507, Springer, 2006. STOCHASTIC

In: Yu. Kabanov, R. Liptser, and J. Stoyanov (editors), From Stochastic Calculus toMathematical Finance: The Shiryaev Festschrift, pp. 433-507, Springer, 2006.

STOCHASTIC DIFFERENTIAL EQUATIONS: A WIENER CHAOSAPPROACH

S. V. LOTOTSKY AND B. L. ROZOVSKII

Abstract. A new method is described for constructing a generalized solution forstochastic differential equations. The method is based on the Cameron-Martin ver-sion of the Wiener Chaos expansion and provides a unified framework for the studyof ordinary and partial differential equations driven by finite- or infinite-dimensionalnoise with either adapted or anticipating input. Existence, uniqueness, regularity,and probabilistic representation of this Wiener Chaos solution is established fora large class of equations. A number of examples are presented to illustrate thegeneral constructions. A detailed analysis is presented for the various forms of thepassive scalar equation and for the first-order Ito stochastic partial differential equa-tion. Applications to nonlinear filtering if diffusion processes and to the stochasticNavier-Stokes equation are also discussed.

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Traditional Solutions of Linear Parabolic Equations. . . . . . . . . . . . . . . . . . . . . . .5

3. White Noise Solutions of Stochastic Parabolic Equations . . . . . . . . . . . . . . . . . 7

4. Generalized Functions on the Wiener Chaos Space . . . . . . . . . . . . . . . . . . . . . . 12

5. The Malliavin Derivative and its Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6. The Wiener Chaos Solution and the Propagator . . . . . . . . . . . . . . . . . . . . . . . . . 17

7. Weighted Wiener Chaos Spaces and S-Transform . . . . . . . . . . . . . . . . . . . . . . . 23

8. General Properties of the Wiener Chaos Solutions . . . . . . . . . . . . . . . . . . . . . . 28

9. Regularity of the Wiener Chaos Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10. Probabilistic Representation of Wiener Chaos Solutions . . . . . . . . . . . . . . . 40

11. Wiener Chaos and Nonlinear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

12. Passive Scalar in a Gaussian Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913. Stochastic Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54

14. First-Order Ito Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2000 Mathematics Subject Classification. Primary 60H15; Secondary 35R60, 60H40.Key words and phrases. Anticipating Equations, Generalized Random Elements, Degenerate Par-abolic Equations, Malliavin Calculus, Passive Scalar Equation, Skorokhod Integral, S-transform,Weighted Spaces.The work of S. V. Lototsky was partially supported by the Sloan Research Fellowship, by the NSFCAREER award DMS-0237724, and by the ARO Grant DAAD19-02-1-0374.The work of B. L. Rozovskii was partially supported by the ARO Grant DAAD19-02-1-0374 andONR Grant N0014-03-1-0027.

1

2 S. V. LOTOTSKY AND B. L. ROZOVSKII

1. Introduction

Consider a stochastic evolution equation

(1.1) du(t) = (Au(t) + f(t))dt + (Mu(t) + g(t))dW (t),

where A andM are differential operators, and W is a noise process on a stochastic basis F =(Ω,F , Ftt≥0,P). Traditionally, this equation is studied under the following assumptions:

(i) The operator A is elliptic, the order of the operator M is at most half the order ofA, and a special parabolicity condition holds.

(ii) The functions f and g are predictable with respect to the filtration Ftt≥0, andthe initial condition is F0-measurable.

(iii) The noise process W is sufficiently regular.

Under these assumptions, there exists a unique predictable solution u of (1.1) so thatu ∈ L2(Ω× (0, T );H) for T > 0 and a suitable function space H (see, for example, Chapter3 of [42]). Moreover, there are examples showing that the parabolicity condition and theregularity of noise are necessary to have a square integrable solution of (1.1).

The objective of the current paper is to study stochastic differential equations of the type(1.1) without making the above assumptions (i)–(iii). We show that, with a suitable def-inition of the solution, solvability of the stochastic equation is essentially equivalent tosolvability of a deterministic evolution equation dv = (Av + ϕ)dt for certain functions ϕ;the operator A does not even have to be elliptic.

Generalized solutions have been introduced and studied for stochastic differential equations,both ordinary and with partial derivatives, and definitions of such solutions relied on variousforms of the Wiener Chaos decomposition. For stochastic ordinary differential equations,Krylov and Veretennikov [20] used multiple Wiener integral expansion to study Ito diffusionswith non-smooth coefficients, and more recently, LeJan and Raimond [22] used a similarapproach in the construction of stochastic flows. Various versions of the Wiener chaosappear in a number of papers on nonlinear filtering and related topics [2, 25, 33, 39, 46,etc.] The book by Holden et al. [12] presents a systematic approach to the stochasticdifferential equations based on the white noise theory. See also [10], [40] and the referencestherein.

For stochastic partial differential equations, most existing constructions of the generalizedsolution rely on various modifications of the Fourier transform in the infinite-dimensionalWiener Chaos space L2(W) = L2(Ω,FW

T ,P). The two main modifications are known as theS-transform [10] and the Hermite transform [12]. The key elements in the development ofthe theory are the spaces of the test functions and the corresponding distributions. Severalconstructions of these spaces were suggested by Hida [10], Kondratiev [17], and Nualartand Rozovskii [38]. Both S- and Hermite transforms establish a bijection between thespace of generalized random elements and a suitable space of analytic functions. Usingthe S-transform, Mikulevicius and Rozovskii [33] studied stochastic parabolic equationswith non-smooth coefficients, while Nualart and Rozovskii [38] and Potthoff et. al [40]constructed generalized solutions for the equations driven by space-time white noise inmore than one spacial dimension. Many other types of equations have been studied, andthe book [12] provides a good overview of literature the corresponding results.

In this paper, generalized solutions of (1.1) are defined in the spaces that are even largerthan Hida or Kondratiev distribution. The Wiener Chaos space is a separable Hilbert

WIENER CHAOS FOR STOCHASTIC EQUATIONS 3

space with a Cameron-Martin basis [3]. The elements of the space with a finite Fourierseries expansion provide the natural collection of test functions D(L2(W)), an analog of thespace D(Rd) of smooth compactly supported functions on Rd. The corresponding spaceof distributions D′(L2(W)) is the collection of generalized random elements represented byformal Fourier series. A generalized solution u = u(t, x) of (1.1) is constructed as an elementof D′(L2(W)) so that the generalized Fourier coefficients satisfy a system of deterministicevolution equations, known as the propagator. If the equation is linear the propagator isa lower-triangular system. We call this solution a Wiener Chaos solution.

The propagator was first introduced by Mikulevicius and Rozovskii in [32], and further stud-ied in [25], as a numerical tool for solving the nonlinear filtering problem. The propagatorcan also be derived for certain nonlinear equations; in particular, it was used in [31, 34, 35]to study the stochastic Navier-Stokes equation.

The propagator approach to defining the solution of (1.1) has two advantages over the S-transform approach. First, the resulting construction is more general: there are equationsfor which the Wiener Chaos solution is not in the domain of the S-transform. Indeed, it isshown in Section 14 that, for certain initial conditions, equation du = uxdWt has a WienerChaos solution for which the S-transform is not defined. On the other hand, by Theorem8.1 below, if the generalized solution of (1.1) can be defined using the S-transform, thenthis solution is also a Wiener Chaos solution. Second, there is no problem of inversion: thepropagator provides a direct approach to studying the properties of Wiener Chaos solutionand computing both the sample trajectories and statistical moments.

Let us emphasize also the following important features of the Wiener Chaos approach:

• The Wiener Chaos solution is a strong solution in the probabilistic sense, that is,it is uniquely determined by the coefficients, free terms, initial condition, and theWiener process.

• The solution exists under minimal regularity conditions on the coefficients in thestochastic part of the equation and no special measurability restriction on the input.

• The Wiener Chaos solution often serves as a convenient first step in the investigationof the traditional solutions or solutions in weighted stochastic Sobolev spaces thatare much smaller then the spaces of Hida or Kondratiev distributions.

To better understand the connection between the Wiener Chaos solution and other notionsof the solution, recall that, traditionally, by a solution of a stochastic equation we understanda random process or field satisfying the equation for almost all elementary outcomes. Thissolution can be either strong or weak in the probabilistic sense.

Probabilistically strong solution is constructed on a prescribed probability space with aspecific noise process. Existence of strong solutions requires certain regularity of the coeffi-cients and the noise in the equation. The tools for constructing strong solutions often comefrom the theory of the corresponding deterministic equations.

Probabilistically weak solution includes not only the solution process but also the stochasticbasis and the noise process. This freedom to choose the probability space and the noiseprocess makes the conditions for existence of weak solutions less restrictive than the similarconditions for strong solutions. Weak solutions can be obtained either by considering thecorresponding martingale problem or by constructing a suitable Hunt process using thetheory of the Dirichlet forms.


There exist equations that have neither weak nor strong solutions in the traditional sense.An example is the bi-linear stochastic heat equation driven by a multiplicative space-timewhite noise in two or more spatial dimensions: the irregular nature of the noise preventsthe existence of a random field that would satisfy the equation for individual elementaryoutcomes. For such equations, the solution must be defined as a generalized random elementsatisfying the equation after the randomness has been averaged out.

White noise theory provides one approach for constructing these generalized solutions. Theapproach is similar to the Fourier integral method for deterministic equations. The whitenoise solution is constructed on a special white noise probability space by inverting anintegral transform; the special structure of the probability space is essential to carry outthe inversion. We can therefore say that the white noise solution extends the notion ofthe probabilistically weak solution. Still, this extension is not a true generalization: whenthe equation satisfies the necessary regularity conditions, the connection between the whitenoise and the traditional weak solution is often not clear.

The Wiener chaos approach provides the means for constructing a generalized solutionon a prescribed probability space. The Wiener Chaos solution is a formal Fourier seriesin the corresponding Cameron-Martin basis. The coefficients in the series are uniquelydetermined by the equation via the propagator system. This representation provides aconvenient way for computing numerically the solution and its statistical moments. As aresult, the Wiener Chaos solution extends the notion of the probabilistically strong solution.Unlike the white noise approach, this is a bona fide extension: when the equation satisfiesthe necessary regularity conditions, the Wiener Chaos solution coincides with the traditionalstrong solution.

After the general discussion of the Wiener Chaos space in Sections 4 and 5, the WienerChaos solution for equation (1.1) and the main properties of the solution are studied inSection 6. Several examples illustrate how the Wiener Chaos solution provides a uniformtreatment of various types of equations: traditional parabolic, non-parabolic, and antici-pating. In particular, for equations with non-predictable input, the Wiener Chaos solutioncorresponds to the Skorohod integral interpretation of the equation. The initial solutionspace D′(W) is too large to provide much of interesting information about the solution.Accordingly, Section 7 discusses various weighted Wiener Chaos spaces. These weightedspaces provide the necessary connection between the Wiener Chaos, white noise, and tra-ditional solutions. This connection is studied in Section 8. In Section 9, the Wiener Chaossolution is constructed for degenerate linear parabolic equations and new regularity resultsare obtained for the solution. Probabilistic representation of the Wiener Chaos solutionis studied in Section 10, where a Feynmann-Kac type formula is derived. Sections 11, 12,13, and 14 discuss the applications of the general results to particular equations: the Zakaifiltering equation, the stochastic transport equation, the stochastic Navier-Stokes equation,and a first-order Ito SPDE.

The following notation will be in force throughout the paper: ∆ is the Laplace operator,Di = ∂/∂xi, i = 1, . . . , d, and summation over the repeated indices is assumed. The spaceof continuous functions is denoted by C, and Hγ

2 , γ ∈ R, is the Sobolev space

f :

∫

R|f(y)|2(1 + |y|2)γdy < ∞

, where f is the Fourier transform of f.


2. Traditional Solutions of Linear Parabolic Equations

Below is a summary of the Hilbert space theory of linear stochastic parabolic equations.The details can be found in the books [41] and [42]; see also [19]. For a Hilbert space X,(·, ·)X and ‖ · ‖X denote the inner product and the norm in X.

Definition 2.1. The triple (V, H, V ′) of Hilbert spaces is called normal if and only if

(1) V → H → V ′ and both embeddings V → H and H → V ′ are dense and continuous;(2) The space V ′ is the dual of V relative to the inner product in H;(3) There exists a constant C > 0 so that |(h, v)H | ≤ C‖v‖V ‖h‖V ′ for all v ∈ V and

h ∈ H.

For example, the Sobolev spaces (H`+γ2 (Rd),H`

2(Rd),H`−γ2 (Rd)), γ > 0, ` ∈ R, form a

normal triple.

Denote by 〈v′, v〉, v′ ∈ V ′, v ∈ V , the duality between V and V ′ relative to the inner productin H. The properties of the normal triple imply that |〈v′, v〉| ≤ C‖v‖V ‖v′‖V ′ , and, if v′ ∈ Hand v ∈ V , then 〈v′, v〉 = (v′, v)H ;

Let F = (Ω,F , Ftt≥0,P) be a stochastic basis with the usual assumptions. In particular,the sigma-algebras F and F0 are P-complete, and the filtration Ftt≥0 is right-continuous;for details, see [23, Definition I.1.1]. We assume that F is rich enough to carry a collectionwk = wk(t), k ≥ 1, t ≥ 0 of independent standard Wiener processes.

Given a normal triple (V, H, V ′) and a family of linear bounded operators A(t) : V → V ′,Mk(t) : V → H, t ∈ [0, T ], consider the following equation:

(2.1) u(t) = u0 +∫ t

0(Au(s) + f(s))ds +

∫ t

0(Mku(s) + gk(s))dwk(s), 0 ≤ t ≤ T,

where T < ∞ is fixed and non-random and the summation convention is in force.

Assume that, for all v ∈ V ,

(2.2)∑

k≥1

‖Mk(t)v‖2H < ∞, t ∈ [0, T ].

The input data u0, f , and gk are chosen so that

(2.3) E

‖u0‖2

H +∫ T

0‖f(t)‖2

V ′dt +∑

k≥1

∫ T

0‖gk(t)‖2

Hdt

< ∞,

u0 is F0-measurable, and the processes f, gk are Ft-adapted, that is, f(t) and each gk(t) areFt-measurable for each t ≥ 0.

Definition 2.2. An Ft-adapted process u ∈ L2(F; L2((0, T );V )) is called a traditional, orsquare-integrable, solution of equation (2.1) if, for every v ∈ V , there exists a measurablesub-set Ω′ of Ω with P(Ω′) = 1, so that, the equality

(2.4) (u(t), v)H = (u0, v)H +∫ t

0〈Au(s) + f(s), v〉ds +

∑

k≥1

(Mku(s) + gk(s), v)Hdwk(s)

holds on Ω′ for all 0 ≤ t ≤ T .

Existence and uniqueness of the traditional solution for (2.1) can be established when theequation is parabolic.


Definition 2.3. Equation (2.1) is called strongly parabolic if there exists a positive num-ber ε and a real number C0 so that, for all v ∈ V and t ∈ [0, T ],

(2.5) 2〈A(t)v, v〉+∑

k≥1

‖M(t)kv‖2H + ε‖v‖2

V ≤ C0‖v‖2H .

Equation (2.1) is called weakly parabolic (or degenerate parabolic) if condition (2.5) holdswith ε = 0.

Theorem 2.4. If (2.3) and (2.5) hold, then there exists a unique traditional solution of(2.1). The solution process u is an element of the space

L2(F; L2((0, T );V ))⋂

L2(F;C((0, T ), H))

and satisfies

E(

sup0<t<T

‖u(t)‖2H +

∫ T

0‖u(t)‖2

V dt

)

≤ C(C0, δ, T )E

‖u0‖2

H +∫ T

0‖f(t)‖2

V ′dt +∑

k≥1

∫ T

0‖gk(t)‖2

Hdt

.

(2.6)

Proof. This follows, for example, from Theorem 3.1.4 in [42]. ¤

A somewhat different solvability result holds for weakly parabolic equations [42, Section3.2].

As an application of Theorem 2.4, consider equationdu(t, x) = (aij(t, x)DiDju(t, x) + bi(t, x)Diu(t, x) + c(t, x)u(t, x) + f(t, x))dt

+ (σik(t, x)Diu(t, x) + νk(t, x)u(t, x) + gk(t, x))dwk(t)(2.7)

with 0 < t ≤ T, x ∈ Rd, and initial condition u(0, x) = u0(x). Assume that

(CL1) The functions aij are bounded and Lipschitz continuous, the functions bi, c, σik,and ν are bounded measurable.

(CL2) There exists a positive number ε > 0 so that

(2aij(x)− σik(x)σjk(x))yiyj ≥ ε|y|2, x, y ∈ Rd, t ∈ [0, T ].

(CL3) There exists a positive number K so that, for all x ∈ Rd,∑

k≥1 |νk(x)|2 ≤ K.

(CL4) The initial condition u0 ∈ L2(Ω; L2(Rd)) is F0-measurable, the processes f ∈L2(Ω × [0, T ];H−1

2 (Rd)) and gk ∈ L2(Ω × [0, T ];L2(Rd)) are Ft-adapted, and∑k≥1

∫ T0 E‖gk‖2

L2(Rd)(t)dt < ∞.

Theorem 2.5. Under assumptions (CL1)–(CL4), equation (2.7) has a unique traditionalsolution

u ∈ L2(F;L2((0, T );H12 (Rd)))

⋂L2(F;C((0, T ), L2(Rd))),

and the solution satisfies

E(

sup0<t<T

‖u‖2L2(Rd)(t) +

∫ T

0‖u‖2

H12 (Rd)(t)dt

)

≤ C(K, ε, T )E

‖u0‖2

L2(Rd) +∫ T

0‖f‖2

H−12 (Rd)

(t)dt+∑

k≥1

∫ T

0‖gk‖2

L2(Rd)(t)dt

.

(2.8)


Proof. Apply Theorem 2.4 in the normal triple(H1

2 (Rd), L2(Rd),H−12 (Rd)); condition (2.5) in this case is equivalent to assumption (CL2).

The details of the proof are in [42, Section 4.1]. ¤

Condition (2.5) essentially means that the deterministic part of the equation dominates thestochastic part. Accordingly, there are two main ways to violate (2.5):

(1) The order of the operatorM is more than half the order of the operatorA. Equationdu = uxdw(t) is an example.

(2) The value of∑

k ‖Mk(t)v‖2H is too large. This value can be either finite, as in

equation du(t, x) = uxx(t, x)dt + 5ux(t, x)dw(t) or infinite, as in equation

(2.9) du(t, x) = ∆u(t, x)dt + σk(x)udwk, σk − CONS in L2(Rd), d ≥ 2.

Indeed, it is shown in [38] that, for equation (2.9), we have∑

k≥1

‖Mk(t)v‖2H = ∞

in every Sobolev space Hγ .

Without condition (2.5), analysis of equation (2.1) requires new technical tools and a dif-ferent notion of solution. The white noise theory provides one possible collection of suchtools.

3. White Noise Solutions of Stochastic Parabolic Equations

The central part of the white noise theory is the mathematical model for the derivative of theBrownian motion. In particular, the Ito integral

∫ t0 f(s)dw(s) is replaced with the integral∫ t

0 f(s)¦ W (s)ds, where W is the white noise process and ¦ is the Wick product. The whitenoise formulation is very different from the Hilbert space approach of the previous section,and requires several new constructions. The book [10] is a general reference about the whitenoise theory, while [12] presents the white noise analysis of stochastic partial differentialequations. Below is the summary of the main definitions and results.

Denote by S = S(R`) the Schwartz space of rapidly decreasing functions and by S ′ = S ′(R`),the Schwartz space of tempered distributions. For the properties of the spaces S and S ′ see[43].

Definition 3.1. The white noise probability space is the triple

S = (S ′,B(S ′), µ),

where B(S ′) is the Borel sigma-algebra of subsets of S ′, and µ is the normalized Gaussianmeasure on B(S ′).

The measure µ is characterized by the property∫

S′e√−1〈ω,ϕ〉dµ(ω) = e

− 12‖ϕ‖2

L2(Rd) ,

where 〈ω, ϕ〉, ω ∈ S ′, ϕ ∈ S, is the duality between S and S ′. Existence of this measurefollows from the Bochner-Minlos theorem [12, Appendix A].


Let ηk, k ≥ 1 be the Hermite basis in L2(R`), consisting of the normalized eigenfunctionsof the operator

(3.1) Λ = −∆ + |x|2, x ∈ R`.

Each ηk is an element of S [12, Section 2.2].

Consider the collection of multi-indices

J1 =

α = (αi, i ≥ 1), αi ∈ 0, 1, 2, . . .,∑

i

αi < ∞

.

The set J1 is countable, and, for every α ∈ J , only finitely many of αi are not equal tozero. For α ∈ J1, write α! =

∏i αi! and define

(3.2) ξα(ω) =1√α!

∏

i

Hαi(〈ω, ηi〉), ω ∈ S ′,

where 〈·, ·〉 is the duality between S and S ′, and

(3.3) Hn(t) = (−1)net2/2 dn

dtne−t2/2

is nth Hermite polynomial. In particular, H1(t) = 1, H1(t) = t, H2(t) = t2 − 1. If, forexample, α = (0, 2, 0, 1, 3, 0, 0, . . .) has three non-zero entries, then

ξα(ω) =H2(〈ω, η2〉)

2!· 〈ω, η4〉 · H3(〈ω, η5〉)

3!.

Theorem 3.2. The collection ξα, α ∈ J1 is an orthonormal basis in L2(S).

Proof. This is a version of the classical result of Cameron and Martin [3]. In this particularform, the result is stated and proved in [12, Theorem 2.2.3]. ¤

By Theorem 3.2, every element ϕ of L2(S) is represented as a Fourier series ϕ =∑

α ϕαξα,where ϕα =

∫S′ ϕ(ω)ξα(ω)dµ, and ‖ϕ‖2

L2(S) =∑

α∈J1|ϕα|2.

For α ∈ J1 and q ∈ R, we write

(2N)qα =∏

j

(2j)qαj .

Definition 3.3. For ρ ∈ [0, 1] and q ≥ 0,

(1) the space (S)ρ,q is the collection of elements ϕ from L2(S) so that

‖ϕ‖2ρ,q =

∑

α∈J1

(α!)ρ(2N)qα|ϕα|2 < ∞;

(2) the space (S)−ρ,−q is the closure of L2(S) relative to the norm

(3.4) ‖ϕ‖2−ρ,−q =

∑

α∈J1

(α!)−ρ(2N)−qα|ϕα|2;

(3) the space (S)ρ is the projective limit of (S)ρ,q as q changes over all non-negativeintegers;

(4) the space (S)−ρ is the inductive limit of (S)−ρ,−q as q changes over all non-negativeintegers.

It follows that


• For each ρ ∈ [0, 1] and q ≥ 0, ((S)ρ,q, L2(S), (S)−ρ,−q) is a normal triple of Hilbertspaces.

• The space (S)ρ is a Frechet space with topology generated by the countable familyof norms ‖ · ‖ρ,n, n = 0, 1, 2, . . ., and ϕ ∈ (S)ρ if and only if ϕ ∈ (S)ρ,q for everyq ≥ 0.

• The space (S)−ρ is the dual of (S)ρ and ϕ ∈ (S)−ρ if and only if ϕ ∈ (S)−ρ,−q forsome q ≥ 0. Every element ϕ from (S)ρ is identified with a formal sum

∑α∈J1

ϕαξα

so that (3.4) holds for some q ≥ 0.• For 0 < ρ < 1,

(S)1 ⊂ (S)ρ ⊂ (S)0 ⊂ L2(S) ⊂ (S)−0 ⊂ (S)−ρ ⊂ (S)−1,

with all inclusions strict.

The spaces (S)0 and (S)1 are known as the spaces of Hida and Kondratiev test functions.The spaces (S)−0 and (S)−1 are known as the spaces of Hida and Kondratiev distributions.Sometimes, the spaces (S)ρ and (S)−ρ, 0 < ρ ≤ 1, go under the name of Kondratiev testfunctions and Kondratiev distributions, respectively.

Let h ∈ S and hk =∫R` h(x)ηk(x)dx. Since the asymptotics of nth eigenvalue of the operator

Λ in (3.1) is n1/d [11, Chapter 21] and Λkh ∈ S for every positive integer k, it follows that

(3.5)∑

k≥1

|hk|2kq < ∞

for every q ∈ R.

For α ∈ J1 and hk as above, write hα =∏

j(hj)αj , and define the stochastic exponential

(3.6) E(h) =∑

α∈J1

hα

√α!

ξα

Lemma 3.4. The stochastic exponential E = E(h), h ∈ S, has the following properties:

• E(h) ∈ (S)ρ, 0 < ρ < 1;• For every q > 0, there exists a δ > 0 so that E(h) ∈ (S)1,q as long as

∑k≥1 |hk|2 < δ.

Proof. Both properties are verified by direct calculation [12, Chapter 2]. ¤

Definition 3.5. The S-transform Sϕ(h) of an element ϕ =∑

α∈J ϕαξα from (S)−ρ is thenumber

(3.7) Sϕ(h) =∑

α∈J1

hα

√α!

ϕα,

where h =∑

k≥1 hkηk ∈ S and hα =∏

j(hj)αj .

The definition implies that if ϕ ∈ (S)−ρ,−q for some q ≥ 0, then Sϕ(h) = 〈ϕ, E(h)〉, where〈·, ·〉 is the duality between (S)ρ,q and (S)−ρ,−q for suitable q. Therefore, if ρ < 1, thenSϕ(h) is well-defined for all h ∈ S, and, if ρ = 1, the Sϕ(h) is well-defined for h withsufficiently small L2(R`) norm. To give a complete characterization of the S-transform, oneadditional construction is necessary.

Let Uρ, 0 ≤ ρ < 1, be the collection of mappings F from S to the complex numbers so that


1. For every h1, h2 ∈ S, the function F (h1+zh2) is an analytic function of the complexvariable z.

2. There exist positive numbers K1,K2 and an integer number n so that, for all h ∈ Sand all complex number z,

|F (zh)| ≤ K1 exp(

K2‖Λnh‖2

1−ρ

L2(Rd)|z| 2

1−ρ

).

For ρ = 1, let U1 be the collection of mappings F from S to the complex numbers so that

1′. There exist ε > 0 and a positive integer n so that, for all h1, h2 ∈ S with‖Λnh1‖L2(R`) < ε, the function of a complex variable z 7→ F (h1 + h2z) is analyticat zero, and

2′. There exists a positive number K so that, for all h ∈ S with ‖Λnh‖L2(R`) < ε,|F (h)| ≤ K.

Two mappings F,G with properties 1′ and 2′ are identified with the same element of U1 ifF = G on an open neighborhood of zero in S.

The following result holds.

Theorem 3.6. For every ρ ∈ [0, 1], the S-transform is a bijection from (S)−ρ to Uρ.

In other words, for every ϕ ∈ (S)−ρ, the S-transform Sϕ is an element of Uρ, and, for everyF ∈ Uρ, there exists a unique ϕ ∈ (S)−ρ so that Sϕ = F . This result is proved in [10] whenρ = 0, and in [17] when ρ = 1.

Definition 3.7. For ϕ and ψ from (S)−ρ, ρ ∈ [0, 1], the Wick product ϕ ¦ ψ is the uniqueelement of (S)−ρ whose S-transform is Sϕ · Sψ.

If S−1 is the inverse S-transform, then

ϕ ¦ ψ = S−1(Sϕ · Sψ),

Note that, by Theorem 3.6, the Wick product is well defined, because the space Uρ, ρ ∈ [0, 1]is closed under the point-wise multiplication. Theorem 3.6 also ensures the correctness ofthe following definition of the white noise.

Definition 3.8. The white noise W on R` is the unique element of (S)0 whose S transformsatisfies SW (h) = h.

Remark 3.9. If g ∈ Lp(S), p > 1, then g ∈ (S)−0 [12, Corollary 2.3.8], and the Fouriertransform

g(h) =∫

S′exp

(√−1〈ω, h〉) g(ω)dµ(ω)

is defined. Direct calculations [12, Section 2.9] show that, for those g,

Sg(√−1h) = g(h) e

12‖h‖2

L2(R`) .

As a result, the Wick product can be interpreted as a convolution on the infinite-dimensionalspace (S)−ρ.

In the study of stochastic parabolic equations, ` = d + 1 so that the generic point fromRd+1 is written as (t, x), t ∈ R, x ∈ Rd. As was mentioned earlier, the terms of the typefdW (t) become f ¦Wdt. The precise connection between the Ito integral and Wick productis discussed, for example, in [12, Section 2.5].


As an example, consider the following equation:

(3.8) ut(t, x) = a(x)uxx(t, x) + b(x)ux(t, x) + ux(t, x) ¦ W (t, x), 0 < t < T, x ∈ R,

with initial condition u(0, x) = u0(x). In (3.8),

(WN1) W is the white noise process on R2.(WN2) The initial condition u0 and the coefficients a, b are bounded and have continuous

bounded derivatives up to second order.(WN3) There exists a positive number ε so that a(x) ≥ ε, x ∈ R.(WN4) The second-order derivative of a is uniformly Holder continuous.

The equivalent Ito formulation of (3.8) is

(3.9) du(t, x) = (a(x)uxx(t, x) + b(x)ux(t, x))dt + ek(x)ux(t, x)dwk(x),

where ek, k ≥ 1 is the Hermite basis in L2(R).

With Mkv = ekvx, we see that condition (2.2) does not hold in any Sobolev space Hγ2 (R).

In fact, no traditional solution exists in any normal triple of Sobolev space. On the otherhand, with a suitable definition of solution, equation (3.8) is solvable in the space (S)−0 ofHida distributions.

Definition 3.10. A mapping u : Rd → (S)−ρ is called weakly differentiable with respect toxi at a point x∗ ∈ R` if and only if there exists a Ui(x∗) ∈ (S)−ρ so that, for all ϕ ∈ (S)ρ,Di〈u(x), ϕ〉|x=x∗ = 〈Ui(x∗), ϕ〉. In that case, we write Ui(x∗) = Diu(x∗).

Definition 3.11. A mapping u from [0, T ]×R to (S)−0 is called a white noise solution of(3.8) if and only if

(1) The weak derivatives ut, ux, and uxx exist, in the sense of Definition 3.10, for all(t, x) ∈ (0, T )× R.

(2) Equality (3.8) holds for all (t, x) ∈ (0, T )× Rd.(3) limt↓0 u(t, x) = u0(x) in the topology of (S)−0.

Theorem 3.12. Under assumptions (WN1)–(WN4), there exists a white noise solution of(3.8). This solution is unique in the class of weakly measurable mappings v from (0, T )×Rto (S)−0, for which there exists a non-negative integer q and a positive number K so that

∫ T

0

∫

R‖v(t, x)‖−0,−qe

−Kx2dxdt < ∞.

Proof. Consider the S-transformed equation

(3.10) Ft(t, x; h) = a(x)Fxx(t, x; h) + b(x)Fx(t, x;h) + Fx(t, x; h)h,

0 < t < T, x ∈ R, h ∈ S(R), with initial condition F (0, x;h) = u0(x). This a deterministicparabolic equation, and one can show, using the probabilistic representation of F , thatF, Ft, Fx, and Fxx belong to U0. Then the inverse S-transform of F is a solution of (3.8),and the uniqueness follows from the uniqueness for equation (3.10). The details of the proofare in [40], where a similar equation is considered for x ∈ Rd. ¤

Even though the initial condition in (3.8) is deterministic, there are no measurability re-strictions on u0 for the white noise solution to exist; see [12] for more details.


With appropriate modifications, the white noise solution can be defined for equations moregeneral than (3.8). The solution F = F (t, x; h) of the corresponding S-transformed equationdetermines the regularity of the white noise solution [12, Section 4.1].

Two main advantages of the white noise approach over the Hilbert space approach are

(1) no need for parabolicity condition;(2) no measurability restrictions on the input data.

Still, there are substantial limitations:

(1) There seems to be little or no connection between the white noise solution and thetraditional solution. While white noise solution can, in principle, be constructed forequation (2.7), this solution will be very different from the traditional solution.

(2) There are no clear ways of computing the solution numerically, even with availablerepresentations of the Feynmann-Kac type [12, Chapter 4].

(3) The white noise solution, being constructed on a special white noise probabilityspace, is weak in the probabilistic sense. Path-wise uniqueness does not apply tosuch solutions because of the ”averaging” nature of the solution spaces.

4. Generalized Functions on the Wiener Chaos Space

The objective of this section is to introduce the space of generalized random elements onan arbitrary stochastic basis.

Let F = (Ω,F , Ftt≥0,P) be a stochastic basis with the usual assumptions and Y , aseparable Hilbert space with inner product (·, ·)Y and an orthonormal basis yk, k ≥ 1.On F and Y , consider a cylindrical Brownian motion W , that is, a family of continuousFt-adapted Gaussian martingales Wy(t), y ∈ Y , so that Wy(0) = 0 and E(Wy1(t)Wy2(s)) =min(t, s)(y1, y2)Y . In particular,

(4.1) wk(t) = Wyk(t), k ≥ 1, t ≥ 0,

are independent standard Wiener processes on F.

Equivalently, instead of the process W , the starting point can be a system of independentstandard Wiener processes wk, k ≥ 1 on F. Then, given a separable Hilbert space Ywith an orthonormal basis yk, k ≥ 1, the corresponding cylindrical Brownian motion Wis defined by

(4.2) Wy(t) =∑

k≥1

(y, yk)Y wk(t).

Fix a non-random T ∈ (0,∞) and denote by FWT the sigma-algebra generated by wk(t), k ≥

1, 0 < t < T . Denote by L2(W) the collection of FWT -measurable square integrable random

variables.

We now review construction of the Cameron-Martin basis in the Hilbert space L2(W).

Let m = mk, k ≥ 1 be an orthonormal basis in L2((0, T )) so that each mk belongs toL∞((0, T )). Define the independent standard Gaussian random variables

ξik =∫ T

0mi(s)dwk(s).


Consider the collection of multi-indices

J =

α = (αki , i, k ≥ 1), αk

i ∈ 0, 1, 2, . . .,∑

i,k

αki < ∞

.

The set J is countable, and, for every α ∈ J , only finitely many of αki are not equal to zero.

The upper and lower indices in αki represent, respectively, the space and time components

of the noise process W . For α ∈ J , define

|α| =∑

i,k

αki , α! =

∏

i,k

αki !,

and

(4.3) ξα =1√α!

∏

i,k

Hαki(ξik),

where Hn is nth Hermite polynomial. For example, if

α =

0 1 0 3 0 0 · · ·2 0 0 0 4 0 · · ·0 0 0 0 0 0 · · ·...

......

......

... · · ·

with four non-zero entries α12 = 1; α1

4 = 3; α21 = 2; α2

5 = 4, then

ξα = ξ2,1 · H3(ξ4,1)√3!

· H2(ξ1,2)√2!

· H4(ξ5,2)√4!

.

There are two main differences between (3.2) and (4.3):

(1) The basis (4.3) is constructed on an arbitrary probability space.(2) In (4.3), there is a clear separation of the time and space components of the noise,

and explicit presence of the time-dependent functions mi facilitates the analysis ofevolution equations.

Definition 4.1. The space L2(W) is called the Wiener Chaos space. The N -th WienerChaos is the linear subspace of L2(W), generated by ξα, |α| = N .

The following is another version of the classical results of Cameron and Martin [3].

Theorem 4.2. The collection Ξ = ξα, α ∈ J is an orthonormal basis in L2(W).

We refer to Ξ as the Cameron-Martin basis in L2(W). By Theorem 4.2, every element v ofL2(W) can be written as

v =∑

α∈Jvαξα,

where vα = E(vξα).

We now define the space D(L2(W)) of test functions and the space D′(L2(W);X) of X-valued generalized random elements.

Definition 4.3.

(1) The space D(L2(W)) is the collection of elements from L2(W) that can be written inthe form

v =∑

α∈Jv

vαξα


for some vα ∈ R and a finite subset Jv of J .(2) A sequence vn converges to v in D(L2(W)) if and only if Jvn ⊆ Jv for all n andlim

n→∞ |vn,α − vα| = 0 for all α.

Definition 4.4. For a linear topological space X define the spaceD′(L2(W);X) of X-valued generalized random elements as the collection of continuous lin-ear maps from the linear topological space D(L2(W)) to X. Similarly, the elements ofD′(L2(W);L1((0, T );X)) are called X-valued generalized random processes.

The element u of D′(L2(W);X) can be identified with a formal Fourier series

u =∑

α∈Juαξα,

where uα ∈ X are the generalized Fourier coefficients of u. For such a series and forv ∈ D(L2(W)), we have

u(v) =∑

α∈Jv

vαuα.

Conversely, for u ∈ D′(L2(W);X), we define the formal Fourier series of u by settinguα = u(ξα). If u ∈ L2(W), then u ∈ D′(L2(W);R) and u(v) = E(uv).

By Definition 4.4, a sequence un, n ≥ 1 converges to u in D′(L2(W);X) if and only ifun(v) converges to u(v) in the topology of X for every v ∈ D(W). In terms of generalizedFourier coefficients, this is equivalent to lim

n→∞un,α = uα in the topology of X for everyα ∈ J .

The construction of the space D′(L2(W);X) can be extended to Hilbert spaces other thanL2(W). Let H be a real separable Hilbert space with an orthonormal basis ek, k ≥ 1.Define the space

D(H) =

v ∈ H : v =∑

k∈Jv

vkek, vk ∈ R, Jv − a finite subset of 1, 2, . . .

.

By definition, vn converges to v in D(H) as n → ∞ if and only if Jvn ⊆ Jv for all n andlim

n→∞ |vn,k − vk| = 0 for all k.

For a linear topological space X, D′(H;X) is the space of continuous linear maps from D(H)to X. An element g of D′(H;X) can be identified with a formal series

∑k≥1 gk⊗ ek so that

gk = g(ek) ∈ X and, for v ∈ D(H), g(v) =∑

k∈Jvgkvk. If X = R and

∑k≥1 g2

k < ∞, theng =

∑k≥1 gkek ∈ H and g(v) = (g, v)H , the inner product in H. The space X is naturally

imbedded into D′(H; X): if u ∈ X, then∑

k≥1 u⊗ ek ∈ D′(H; X).

A sequence gn =∑

k≥1 gn,k ⊗ ek, n ≥ 1, converges to g =∑

k≥1 gk ⊗ ek in D′(H; X) if andonly if, for every k ≥ 1, lim

n→∞ gn,k = gk in the topology of X.

A collection Lk, k ≥ 1 of linear operators from X1 to X2 naturally defines a linearoperator L from D′(H;X1) to D′(H; X2):

L∑

k≥1

gk ⊗ ek

=

∑

k≥1

Lk(gk)⊗ ek.

Similarly, a linear operator L : D′(H; X1) → D′(H; X2) can be identified with a collectionLk, k ≥ 1 of linear operators from X1 to X2 by setting Lk(u) = L(u⊗ ek). Introduction


of spaces D′(H; X) and the corresponding operators makes it possible to avoid conditionsof the type (2.2).

5. The Malliavin Derivative and its Adjoint

In this section, we define an analog of the Ito stochastic integral for generalized randomprocesses.

All notations from the previous section will remain in force. In particular, Y is a separableHilbert space with a fixed orthonormal basis yk, k ≥ 1, and Ξ = ξα, α ∈ J , theCameron-Martin basis in L2(W) defined in (4.3).

We start with a brief review of the Malliavin calculus [37].

The Malliavin derivative D is a continuous linear operator from

(5.1) L12(W) =

u ∈ L2(W) :

∑

α∈J|α|u2

α < ∞

to L2 (W; (L2((0, T ))× Y )). In particular,

(5.2) (Dξα)(t) =∑

i,k

√αk

i ξα−(i,k)mi(t)yk,

where α−(i, k) is the multi-index with the components(α−(i, k)

)l

j=

max(αk

i − 1, 0), if i = j and k = l,αl

j , otherwise.

Note that, for each t ∈ [0, T ], Dξα(t) ∈ D(L2(W)× Y ). Using (5.2), we extend the operatorD by linearity to the space D′(L2(W)):

D

(∑

α∈Juαξα

)=

∑

α∈J

uα

∑

i,k

√αk

i ξα−(i,k)mi(t)yk

.

For the sake of completeness and to justify further definitions, let us establish connectionbetween the Malliavin derivative and the stochastic Ito integral.

If u is an FWt -adapted process from L2 (W;L2((0, T );Y )), then u(t) =

∑k≥1 uk(t)yk, where

the random variable uk(t) is FWt -measurable for each t and k, and

∑

k≥1

∫ T

0E|uk(t)|2dt < ∞.

We define the stochastic Ito integral

(5.3) U(t) =∫ t

0(u(s), dW (s))Y =

∑

k≥1

∫ t

0uk(s)dwk(s).

Note that U(t) is FWt -measurable and E|U(t)|2 =

∑k≥1

∫ t0 E|uk(s)|2ds.

The next result establishes a connection between the Malliavin derivative and the stochasticIto integral.


Lemma 5.1. Suppose that u is an FWt -adapted process from

L2 (W;L2((0, T );Y )), and define the process U according to (5.3). Then, for every 0 < t ≤T and α ∈ J ,

(5.4) E(U(t)ξα) = E∫ t

0(u(s), (Dξα)(s))Y ds.

Proof. Define ξα(t) = E(ξα|FWt ). It is known (see [33] or Remark 8.3 below) that

(5.5) dξα(t) =∑

i,k

√αk

i ξα−(i,k)(t)mi(t)dwk(t).

Due to FWt -measurability of uk(t), we have

(5.6) uk,α(t) = E(uk(t)E(ξα|FW

t ))

= E(uk(t)ξα(t)).

The definition of U implies dU(t) =∑

k≥1 uk(t)dwk(t), so that, by (5.5), (5.6), and the Itoformula,

(5.7) Uα(t) = E(U(t)ξα) =∫ t

0

∑

i,k

√αk

i uk,α−(i,k)(s)mi(s)ds.

Together with (5.2), the last equality implies (5.4). Lemma 5.1 is proved. ¤

Note that the coefficients uk,α of u ∈ L2(W;L2((0, T );H)) belong to L2((0, T )). We there-fore define uk,α,i =

∫ T0 uk,α(t)mi(t)dt. Then, by (5.7),

(5.8) Uα(T ) =∑

i,k

√αk

i uk,α−(i,k),i.

Since U(T ) =∑

α∈J Uα(T )ξα, we shift the summation index in (5.8) and conclude that

(5.9) U(T ) =∑

α∈J

∑

i,k

√αk

i + 1uk,α,iξα+(i,k),

where

(5.10)(α+(i, k)

)l

j=

αk

i + 1, if i = j and k = l,αl

j , otherwise.

As a result, U(T ) = δ(u), where δ is the adjoint of the Malliavin derivative, also known asthe Skorokhod integral; see [37] or [38] for details.

Lemma 5.1 suggests the following definition. For an FWt -adapted process u from

L2 (W;L2((0, T ))), let D∗ku be the FWt -adapted process from L2 (W; L2((0, T ))) so that

(5.11) (D∗ku)α(t) =∫ t

0

∑

i

√αk

i uα−(i,k)(s)mi(s)ds.

If u ∈ L2 (W; L2((0, T );Y )) is FWt -adapted, then u is in the domain of the operator δ and

δ(uI(s < t)) =∑

k≥1(D∗kuk)(t).

We now extend the operators D∗k to the generalized random processes. Let X be a Banachspace with norm ‖ · ‖X .


Definition 5.2. If u is an X-valued generalized random process, then D∗ku is the X-valuedgeneralized random process so that

(5.12) (D∗ku)α(t) =∑

i

∫ t

0uα−(i,k)(s)

√αk

i mi(s)ds.

If g ∈ D′(Y ;D′ (L2(W);L1((0, T );X))

), then D∗g is the X-valued generalized random

process so that, for g =∑

k≥1 gk ⊗ yk, gk ∈ D′(L2(W);L1((0, T );X)),

(5.13) (D∗g)α(t) =∑

k

(D∗kgk)α(t) =∑

i,k

∫ t

0gk,α−(i,k)(s)

√αk

i mi(s)ds.

Using (5.2), we get a generalization of equality (5.4):

(5.14) (D∗g)α(t) =∫ t

0g(Dξα(s))(s)ds.

Indeed, by linearity,

gk

(√αk

i mi(s)ξα−(i,k)

)(s) =

√αk

i mi(s)gk,α−(i,k))(s).

Theorem 5.3. If T < ∞, then D∗k and D∗ are continuous linear operators.

Proof. It is enough to show that, if u, un ∈ D′(L2(FW

T );L1((0, T );X))

andlimn→∞ ‖uα − un,α‖L1((0,T );X) = 0 for every α ∈ J , then, for every k ≥ 1 and α ∈ J ,limn→∞ ‖(D∗ku)α − (D∗kun)α‖L1((0,T );X) = 0.

Using (5.12), we find

‖(D∗ku)α − (D∗kun)α‖X(t) ≤∑

i

∫ T

0

√αk

i ‖uα−(i,k) − un,α−(i,k)‖X(s)|mi(s)|ds.

Note that the sum contains finitely many terms. By assumption, |mi(t)| ≤ Ci, and so

‖(D∗ku)α − (D∗kun)α‖L1((0,T );X)≤C(α)∑

i

√αk

i ‖uα−(i,k) − un,α−(i,k)‖L1((0,T );X).

Theorem 5.3 is proved. ¤

6. The Wiener Chaos Solution and the Propagator

In this section we build on the ideas from [25] to introduce the Wiener Chaos solution andthe corresponding propagator for a general stochastic evolution equation. The notationsfrom Sections 4 and 5 will remain in force. It will be convenient to interpret the cylindricalBrownian motion W as a collection wk, k ≥ 1 of independent standard Wiener processes.As before, T ∈ (0,∞) is fixed and non-random. Introduce the following objects:

• The Banach spaces A, X, and U so that U ⊆ X.• Linear operators

A : L1((0, T );A) → L1((0, T );X) andMk : L1((0, T );A) → L1((0, T );X).

• Generalized random processes f ∈ D′ (L2(W);L1((0, T );X)) andgk ∈ D′ (L2(W);L1((0, T );X)) .


• The initial condition u0 ∈ D′ (L2(W);U).

Consider the deterministic equation

(6.1) v(t) = v0 +∫ t

0(Av)(s)ds +

∫ t

0ϕ(s)ds,

where v0 ∈ U and ϕ ∈ L1((0, T );X).

Definition 6.1. A function v is called a w(A, X) solution of (6.1) if and only if v ∈L1((0, T );A) and equality (6.1) holds in the space L1((0, T );A).

Definition 6.2. An A-valued generalized random process u is called a w(A, X) WienerChaos solution of the stochastic differential equation

(6.2) du(t) = (Au(t) + f(t))dt + (Mku(t) + gk(t))dwk(t), 0 < t ≤ T, u|t=0 = u0,

if and only if the equality

(6.3) u(t) = u0 +∫ t

0(Au + f)(s)ds +

∑

k≥1

(D∗k(Mku + gk))(t)

holds in D′ (L2(W);L1((0, T );X)).

Sometimes, to stress the dependence of the Wiener Chaos solution on the terminal time T ,the notation wT (A,X) will be used.

Equalities (6.3) (5.13) mean that, for every α ∈ J , the generalized Fourier coefficient uα ofu satisfies

(6.4) uα(t) = u0,α +∫ t

0(Au + f)α(s)ds +

∫ t

0

∑

i,k

√αk

i (Mku + gk)α−(i,k)(s)mi(s)ds.

Definition 6.3. System (6.4) is called the propagator for equation (6.2).

The propagator is a lower triangular system. Indeed, If α = (0), that is, |α| = 0, then thecorresponding equation in (6.4) becomes

(6.5) u(0)(t) = u0,(0) +∫ t

0(Au(0)(s) + f(0)(s))ds.

If α = (j`), that is, α`j = 1 for some fixed j and ` and αk

i = 0 for all other i, k ≥ 1, then thecorresponding equation in (6.4) becomes

u(j`)(t) = u0,(j`) +∫ t

0(Au(j`)(s) + f(j`)(s))ds

+∫ t

0(Mku(0)(s) + g`,(0)(s))mj(s)ds.

(6.6)

Continuing in this way, we conclude that (6.4) can be solved by induction on |α| as long asthe corresponding deterministic equation (6.1) is solvable. The precise result is as follows.

Theorem 6.4. If, for every v0 ∈ U and ϕ ∈ L1((0, T );X), equation (6.1) has a uniquew(A,X) solution v(t) = V (t, v0, ϕ), then equation (6.2) has a unique w(A,X) Wiener Chaos


solution so that

uα(t) = V (t, u0,α, fα) +∑

i,k

√αk

i V (t, 0,miMkuα−(i,k))

+∑

i,k

√αk

i V (t, 0,migk,α−(i,k)).(6.7)

Proof. Using the assumptions of the theorem and linearity, we conclude that (6.7) is theunique solution of (6.4). ¤

To derive a more explicit formula for uα, we need some additional constructions. For everymulti-index α with |α| = n, define the characteristic set Kα of α so that

Kα = (iα1 , kα1 ), . . . , (iαn, kα

n),iα1 ≤ iα2 ≤ . . . ≤ iαn, and if iαj = iαj+1, then kα

j ≤ kαj+1. The first pair (iα1 , kα

1 ) in Kα is theposition numbers of the first nonzero element of α. The second pair is the same as thefirst if the first nonzero element of α is greater than one; otherwise, the second pair is theposition numbers of the second nonzero element of α and so on. As a result, if αk

i > 0, thenexactly αk

i pairs in Kα are equal to (i, k). For example, if

α =

0 1 0 2 3 0 0 · · ·1 2 0 0 0 1 0 · · ·0 0 0 0 0 0 0 · · ·...

......

......

...... · · ·

with nonzero elements

α21 = α1

2 = α61 = 1, α2

2 = α14 = 2, α1

5 = 3,

then the characteristic set is

Kα =(1, 2), (2, 1), (2, 2), (2, 2), (4, 1), (4, 1), (5, 1), (5, 1), (5, 1), (6, 2).Theorem 6.5. Assume that

(1) for every v0 ∈ U and ϕ ∈ L1((0, T );X), equation (6.1) has a unique w(A,X)solution v(t) = V (t, v0, ϕ),

(2) the input data in (6.4) satisfy gk = 0 and fα = u0,α = 0 if |α| > 0.

Let u(0)(t) = V (t, u0, 0) be the solution of (6.4) for |α| = 0. For α ∈ J with |α| = n ≥ 1and the characteristic set Kα, define functions Fn = Fn(t;α) by induction as follows:

F 1(t; α) = V (t, 0,miMku(0)) if Kα = (i, k);

Fn(t; α) =n∑

j=1

V (t, 0,mijMkjFn−1(·; α−(ij , kj)))

if Kα = (i1, k1), . . . , (in, kn).

(6.8)

Then

(6.9) uα(t) =1√α!

Fn(t; α).

Proof. If |α| = 1, then representation (6.9) follows from (6.6). For |α| > 1, observe that


• If uα(t) =√

α!uα and |α| ≥ 1, then (6.4) implies

u(t) =∫ t

0Auα(s)ds +

∑

i,k

∫ t

0αk

i mi(s)Mkuα−(i,k)(s)ds.

• If Kα = (i1, k1), . . . , (in, kn), then, for every j = 1, . . . , n, the characteristic setKα−(ij ,kj) of α−(ij , kj) is obtained from Kα by removing the pair (ij , kj).

• By the definition of the characteristic set,

∑

i,k

αki mi(s)Mkuα−(i,k)(s) =

n∑

j=1

mij (s)Mkj uα−(ij ,kj)(s).

As a result, representation (6.9) follows by induction on |α| using (6.7):if |α| = n > 1, then

uα(t) =n∑

j=1

V (t, 0,mijMkj uα−(ij ,kj))

=n∑

j=1

V (t, 0,mijMkjF(n−1)(·; α−(ij , kj)) = Fn(t; α).

(6.10)

Theorem 6.5 is proved. ¤

Corollary 6.6. Assume that the operator A is a generator of a strongly continuous semi-group Φ = Φt,s, t ≥ s ≥ 0, in some Hilbert space H so that A ⊂ H, each Mk is a boundedoperator from A to H, and the solution V (t, 0, ϕ) of equation (6.1) is written as

(6.11) V (t, 0, ϕ) =∫ T

0Φt,sϕ(s)ds, ϕ ∈ L2((0, T );H)).

Denote by Pn the permutation group of 1, . . . , n. If u(0) ∈ L2((0, T );H)), then, for |α| =n > 1 with the characteristic set Kα = (i1, k1), . . . , (in, kn), representation (6.9) becomes

uα(t) =1√α!

∑

σ∈Pn

∫ t

0

∫ sn

0. . .

∫ s2

0

Φt,snMkσ(n)· · ·Φs2,s1Mkσ(1)

u(0)(s1)miσ(n)(sn) · · ·miσ(1)

(s1)ds1 . . . dsn.

(6.12)

Also,∑

|α|=n

uα(t)ξα =∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0

Φt,snMkn · · ·Φs2,s1

(Mk1u(0) + gk1(s1))dwk1(s1) · · · dwkn(sn), n ≥ 1,

(6.13)

and, for every Hilbert space X, the following energy equality holds:

∑

|α|=n

‖uα(t)‖2X =

∞∑

k1,...,kn=1

∫ t

0

∫ sn

0. . .

∫ s2

0

‖Φt,snMkn · · ·Φs2,s1Mk1u(0)(s1)‖2Xds1 . . . dsn;

(6.14)

both sides in the last equality can be infinite. For n = 1, formulas (6.12) and (6.14) become

(6.15) u(ik)(t) =∫ t

0Φt,sMku(0)(s) mi(s)ds;


(6.16)∑

|α|=1

‖uα(t)‖2X =

∞∑

k=1

∫ t

0‖Φt,sMku(0)(s)‖2

Xds.

Proof. Using the semi-group representation (6.11), we conclude that (6.12) is just an ex-panded version of (6.9).

Since mi, i ≥ 1 is an orthonormal basis in L2(0, T ), equality (6.16) follows from (6.15) andthe Parcevall identity. Similarly, equality (6.14) will follow from (6.12) after an applicationof an appropriate Parcevall’s identity.

To carry out the necessary arguments when |α| > 1, denote by J1 the collection of one-dimensional multi-indices β = (β1, β2, . . .) so that each βi is a non-negative integer and|β| =

∑i≥1 βi < ∞. Given a β ∈ J1 with |β| = n, we define Kβ = i1, . . . , in, the

characteristic set of β and the function

(6.17) Eβ(s1, . . . , sn) =1√β!n!

∑

σ∈Pn

mi1(sσ(1)) · · ·min(sσ(n)).

By construction, the collection Eβ, β ∈ J1, |β| = n is an orthonormal basis in the sub-space of symmetric functions in L2((0, T )n; X).

Next, we re-write (6.12) in a symmetrized form. To make the notations shorter, denoteby s(n) the ordered set (s1, . . . , sn) and write dsn = ds1 . . . dsn. Fix t ∈ (0, T ] and theset k(n) = k1, . . . , kn of the second components of the characteristic set Kα. Define thesymmetric function

G(t, k(n); s(n))

=1√n!

∑

σ∈Pn

Φt,sσ(n)Mkn · · ·Φsσ(2),sσ(1)

Mk1u(0)(sσ(1))1sσ(1)<···<sσ(n)<t(s(n)).(6.18)

Then (6.12) becomes

(6.19) uα(t) =∫

[0,T ]nG(t, k(n); s(n))Eβ(α)(s

(n))dsn,

where the multi-indices α and β(α) are related via their characteristic sets: if

Kα = (i1, k1), . . . , (in, kn),then

Kβ(α) = i1, . . . , in.Equality (6.19) means that, for fixed k(n), the function uα is a Fourier coefficient of thesymmetric function G(t, k(n); s(n)) in the space L2((0, T )n; X). Parcevall’s identity andsummation over all possible k(n) yield

∑

|α|=n

‖uα(t)‖2X =

1n!

∞∑

k1,...,kn=1

∫

[0,T ]n‖G(t, k(n); s(n))‖2

Xdsn,

which, due to (6.18), is the same as (6.14).

To prove equality (6.13), relating the Cameron-Martin and multiple Ito integral expansionsof the solution, we use the following result [13, Theorem 3.1]:

ξα =1√α!

∫ T

0

∫ sn

0· · ·

∫ s2

0Eβ(α)(s

(n))dwk1(s1) · · · dwkn(sn);


see also [37, pp. 12–13]. Since the collection of all Eβ is an orthonormal basis, equality(6.13) follows from (6.19) after summation over al k1, . . . , kn.

Corollary 6.6 is proved. ¤

We now present several examples to illustrate the general results.

Example 6.7. Consider the following equation:

(6.20) du(t, x) = (auxx(t, x) + f(t, x))dt + (σux(t, x) + g(t, x))dw(t), t > 0, x ∈ R,

where a > 0, σ ∈ R, f ∈ L2((0, T );H−12 (R)), g ∈ L2((0, T );L2(R)), and u|t=0 = u0 ∈

L2(R). By Theorem 2.5, if σ2 < 2a, then equation (6.20) has a unique traditional solutionu ∈ L2

(W;L2((0, T );H1

2 (R))).

By FWt -measurability of u(t), we have

E(u(t)ξα) = E(u(t)E(ξα|FWt )).

Using the relation (5.5) and the Ito formula, we find that uα satisfy

duα = a(uα)xxdt +∑

i

√αiσ(uα−(i))xmi(t)dt,

which is precisely the propagator for equation (6.20). In other words, if 2a > σ2, then thetraditional solution of (6.20) coincides with the Wiener Chaos solution.

On the other hand, the heat equation

v(t, x) = v0(x) +∫ t

0vxx(s, x)ds +

∫ t

0ϕ(s, x)ds, v0 ∈ L2(R)

with ϕ ∈ L2((0, T );H−12 (R)) has a unique w(H1

2 (R), H−12 (R)) solution. Therefore, by Theo-

rem 6.4, the unique w(H12 (R), H−1

2 (R)) Wiener Chaos solution of (6.20) exists for all σ ∈ R.

In the next example, the equation, although not parabolic, can be solved explicitly.

Example 6.8. Consider the following equation:

(6.21) du(t, x) = ux(t, x)dw(t), t > 0, x ∈ R; u(0, x) = x.

Clearly, u(t, x) = x + w(t) satisfies (6.21).

To find the Wiener Chaos solution of (6.21), note that, with one-dimensional Wiener process,αk

i = αi, and the propagator in this case becomes

uα(t, x) = xI(|α| = 0) +∫ t

0

∑

i

√αi(uα−(i)(s, x))xmi(s)ds.

Then uα = 0 if |α| > 1, and

(6.22) u(t, x) = x +∑

i≥1

ξi

∫ t

0mi(s)ds = x + w(t).

Even though Theorem 6.4 does not apply, the above arguments show that u(t, x) = x+w(t)is the unique w(A,X) Wiener Chaos solution of (6.21) for suitable spaces A and X, forexample,

X =

f :∫

R(1 + x2)−2f2(x)dx < ∞

and A = f : f, f ′ ∈ X.

Section 14 provides a more detailed analysis of equation (6.21).


If equation (6.2) is anticipating, that is, the initial condition is not deterministic and/orthe free terms f, g are not FW

t -adapted, then the Wiener Chaos solution generalizes theSkorohod integral interpretation of the equation.

Example 6.9. Consider the equation

(6.23) du(t, x) =12uxx(t, x)dt + ux(t, x)dw(t), t ∈ (0, T ], x ∈ R,

with initial condition u(0, x) = x2w(T ). Since w(T ) =√

Tξ1, we find

(6.24) (uα)t(t, x) =12(uα)xx(t, x) +

∑

i

√αimi(t)(uα−(i))x(t, x)

with initial condition uα(0, x) =√

Tx2I(|α| = 1, α1 = 1). By Theorem 6.4, there exists aunique w(A,X) Wiener Chaos solution of (6.23) for suitable spaces A and X. For example,we can take

X =

f :∫

R(1 + x2)−8f2(x)dx < ∞

and A = f : f, f ′, f ′′ ∈ X.

System (6.24) can be solved explicitly. Indeed, uα ≡ 0 if |α| = 0 or |α| > 3 or if α1 = 0.Otherwise, writing Mi(t) =

∫ t0 mi(s)ds, we find:

uα(t, x) = (t + x2)√

T , if |α| = 1, α1 = 1;

uα(t, x) = 2√

2 xt, if |α| = 2, α1 = 2;

uα(t, x) = 2√

T xMi(t), if |α| = 2, α1 = αi = 1, 1 < i;

uα(t, x) =

√6T

t2, if |α| = 3, α1 = 3;

uα(t, x) = 2√

2T M1(t)Mi(t), if |α| = 3, α1 = 2, αi = 1, 1 < i;

uα(t, x) =√

2T M2i (t), if |α| = 3, α1 = 1, αi = 2, 1 < i;

uα(t, x) = 2√

T Mi(t)Mj(t), if |α| = 3, α1 = αi = αj = 1, 1 < i < j.

Then

(6.25) u(t, x) =∑

α∈Juαξα = w(T )w2(t)− 2tw(t) + 2(W (T )w(t)− t)x + x2w(T )

is the Wiener Chaos solution of (6.23). It can be verified using the properties of the Skorohodintegral [37] that the function u defined by (6.25) satisfies

u(t, x) = x2w(T ) +12

∫ t

0uxx(s, x)ds +

∫ t

0ux(s, x)dw(s), t ∈ [0, T ], x ∈ R,

where the stochastic integral is in the sense of Skorohod.

7. Weighted Wiener Chaos Spaces and S-Transform

The space D′(L2(W);X) is too big to provide any reasonable information about regularity ofthe Wiener Chaos solution. Introduction of weighted Wiener chaos spaces makes it possibleto resolve this difficulty.

As before, let Ξ = ξα, α ∈ J be the Cameron-Martin basis in L2(W), and D(L2(W);X),the collection of finite linear combinations of ξα with coefficients in a Banach space X.


Definition 7.1. Given a collection rα, α ∈ J of positive numbers, the space RL2(W; X)is the closure of D(L2(W);X) with respect to the norm

‖v‖2RL2(W;X) :=

∑

α∈Jr2α‖vα‖2

X .

The operator R defined by (Rv)α := rαvα is a linear homeomorphism from RL2(W;X) toL2(W;X).

There are several special choices of the weight sequence R = rα, α ∈ J and specialnotations for the corresponding weighted Wiener chaos spaces.

• If Q = q1, q2, . . . is a sequence of positive numbers, define

qα =∏

i,k

qαk

ik .

The operator R, corresponding to rα = qα, is denotes by Q. The space QL2(W; X)is denoted by L2,Q(W; X) and is called a Q-weighted Wiener chaos space. Thesignificance of this choice of weights will be explained shortly (see, in particular,Proposition 7.4).

• Ifr2α = (α!)ρ

∏

i,k

(2ik)γαki , ρ, γ ∈ R,

then the corresponding space RL2(W; X) is denoted by (S)ρ,γ(X). As always, theargument X will be omitted if X = R. Note the analogy with Definition 3.3.

The structure of weights in the spaces L2,Q and (S)ρ,γ is different, and in general thesetwo classes of spaces are not related. There exist generalized random elements that belongto some L2,Q(W; X), but do not belong to any (S)ρ,γ(X). For example, u =

∑k≥1 ek2

ξ1,k

belongs to L2,Q(W) with qk = e−2k2, but to no (S)ρ,γ , because the sum

∑k≥1 e2k2

(k!)ρ(2k)γ

diverges for every ρ, γ ∈ R. Similarly, there exist generalized random elements that belongto some (S)ρ,γ(X), but to no L2,Q(W; X). For example, u =

∑n≥1

√n!ξ(n), where (n) is the

multi-index with α11 = n and αk

i = 0 elsewhere, belongs to (S)−1,−1, but does not belong toany L2,Q(W), because the sum

∑n≥1 qnn! diverges for every q > 0.

The next result is the space-time analog of Proposition 2.3.3 in [12].

Proposition 7.2. The sum ∑

α∈J

∏

i,k≥1

(2ik)−γαki

converges if and only if γ > 1.

Proof. Note that

(7.1)∑

α∈J

∏

i,k≥1

(2ik)−γαki =

∏

i,k≥1

∑

n≥0

((2ik)−γ)n

=

∏

i,k

1(1− (2ik)−γ)

, γ > 0

The infinite product on the right of (7.1) converges if and only if each of the sums∑

i≥1 i−γ ,∑k≥1 k−γ converges, that is, if an only if γ > 1. ¤

Corollary 7.3. For every u ∈ D′(W; X), there exists an operator R so thatRu ∈ L2(W; X).


Proof. Define

r2α =

11 + ‖uα‖2

X

∏

i,k≥1

(2ik)−2αki .

Then

‖Ru‖2L2(W;X) =

∑

α∈J

‖uα‖2X

1 + ‖uα‖2X

∏

i,k≥1

(2ik)−2αki ≤

∑

α∈J

∏

i,k≥1

(2ik)−2αki < ∞.

¤

The importance of the operator Q in the study of stochastic equations is due to the factthat the operator R maps a Wiener Chaos solution to a Wiener Chaos solution if and onlyR = Q for some sequence Q. Indeed, direct calculations show that the functions uα, α ∈ J ,satisfy the propagator (6.4) if and only if vα = (Ru)α satisfy

vα(t) = (Ru0)α +∫ t

0(Av +Rf)α(s)ds

+∫ t

0

∑

i,k

√αk

i

ρα

ρα−(i,k)(MkRu +Rgk)α−(i,k)(s)mi(s)ds.

(7.2)

Therefore, the operator R preserves the structure of the propagator if and only ifρα

ρα−(i,k)= qk,

that is, ρα = qα for some sequence Q.

Below is the summary of the main properties of the operator Q.

Proposition 7.4.

(1) If qk ≤ q < 1 for all k ≥ 1, then L2,Q(W) ⊂ (S)0,−γ for some γ > 0.(2) If qk ≥ q > 1 for all k, then L2,Q(W) ⊂ Ln

2 (W) for all n ≥ 1, that is, the elementsof L2,Q(W) are infinitely differentiable in the Malliavin sense.

(3) If u ∈ L2,Q(W;X) with generalized Fourier coefficients uα satisfying the propaga-tor (6.4), and v = Qu, then the corresponding system for the generalized Fouriercoefficients of v is

vα(t) = (Qu0)α +∫ t

0(Av +Qf)α(s)ds

+∫ t

0

∑

i,k

√αk

i (Mkv +Qgk)α−(i,k)(s)qkmi(s)ds.(7.3)

(4) The function u is a Wiener Chaos solution of

(7.4) u(t) = u0 +∫ t

0(Au(s) + f(s))dt +

∫ t

0(Mu(s) + g(s), dW (s))Y

if and only if v = Qu is a Wiener Chaos solution of

(7.5) v(t) = (Qu)0 +∫ t

0(Av(s) +Qf(s))dt +

∫ t

0(Mv(s) +Qg(s), dWQ(s))Y ,

where, for h ∈ Y , WQh (t) =

∑k≥1(h, yk)Y qkwk(t).


The following examples demonstrate how the operator Q helps with the analysis of variousstochastic evolution equations.

Example 7.5. Consider the w(H12 (R),H−1

2 (R)) Wiener Chaos solution u of equation

(7.6) du(t, x) = (auxx(t, x) + f(t, x))dt + σux(t, x)dw(t), 0 < t ≤ T, x ∈ R,

with f ∈ L2(Ω × (0, T );H−12 (R)), g ∈ L2(Ω × (0, T );L2(R)), and u|t=0 = u0 ∈ L2(R).

Assume that σ > 0 and define the sequence Q so that qk = q for all k ≥ 1 and q <√

2a/σ.By Theorem 2.5, equation

dv = (avxx + f)dt + (qσux + g)dw

with v|t=0 = u0, has a unique traditional solution

v ∈ L2

(W; L2((0, T );H1

2 (R))) ⋂

L2 (W;C((0, T );L2(R))) .

By Proposition 7.4, the w(H12 (R),H−1

2 (R)) Wiener Chaos solution u of equation (7.6) sat-isfies u = Q−1v and

u ∈ L2,Q

(W; L2((0, T );H1

2 (R))) ⋂

L2,Q (W;C((0, T );L2(R))) .

Note that if equation (7.6) is strongly parabolic, that is, 2a > σ2, then the weight q can betaken bigger than one, and, according to the first statement of Proposition 7.4, regularityof the solution is better than the one guaranteed by Theorem 2.5.

Example 7.6. The Wiener Chaos solutions can be constructed for stochastic ordinarydifferential equations. Consider, for example,

(7.7) u(t) = 1 +∫ t

0

∑

k≥1

u(s)dwk(s),

which clearly does not have a traditional solution. On the other hand, the unique w(R,R)Wiener Chaos solution of this equation belongs to L2,Q (W; L2((0, T )) for every Q satisfying∑

k q2k < ∞. Indeed, for (7.7), equation (7.5) becomes

v(t) = 1 +∫ t

0

∑

k

v(s)qkdwk(s).

If∑

k q2k < ∞, then the traditional solution of this equation exists and belongs to

L2 (W;L2((0, T ))).

There exist equations for which the Wiener Chaos solution does not belong to any weightedWiener chaos space L2,Q. An example is given below in Section 14.

To define the S-transform, consider the following analog of the stochastic exponential (3.6).

Lemma 7.7. If h ∈ D (L2((0, T );Y )) and

E(h) = exp(∫ T

0(h(t), dW (t))Y − 1

2

∫ T

0‖h(t)‖2

Y dt

),

then

• E(h) ∈ L2,Q(W) for every sequence Q.• E(h) ∈ (S)ρ,γ for 0 ≤ ρ < 1 and γ ≥ 0.• E(h) ∈ (S)1,γ, γ ≥ 0, as long as ‖h‖2

L2((0,T );Y ) is sufficiently small.


Proof. Recall that, if h ∈ D(L2((0, T );Y )), then h(t) =∑

i,k∈Ihhk,imi(t)yk, where Ih is a

finite set. Direct computations show that

E(h) =∏

i,k

∑

n≥0

Hn(ξik)n!

(hk,i)n

=

∑

α∈J

hα

√α!

ξα

where hα =∏

i,k hαk

ik,i. In particular,

(7.8) (E(h))α =hα

√α!

.

Consequently, for every sequence Q of positive numbers,

(7.9) ‖E(h)‖2L2,Q(W) = exp

∑

i,k∈Ih

h2k,iq

2k

< ∞.

Similarly, for 0 ≤ ρ < 1 and γ ≥ 0,

(7.10) ‖E(h)‖2(S)ρ,γ

=∑

α∈J

∏

i,k

((2ik)γhk,i)2αki

(αki !)1−ρ

=∏

i,k∈Ih

∑

n≥0

((2ik)γhk,i)2n

(n!)1−ρ

< ∞,

and, for ρ = 1,

(7.11) ‖E(h)‖2(S)1,γ

=∑

α∈J

∏

i,k

((2ik)γhk,i)2αki =

∏

i,k∈Ih

∑

n≥0

((2ik)γhk,i)2n

< ∞,

if 2(max(m,n)∈Ih)(mn)γ

)∑i,k h2

k,i < 1. Lemma 7.7 is proved. ¤

Remark 7.8. It is well-known (see, for example, [24, Proof of Theorem 5.5]) that thefamily E(h), h ∈ D (L2((0, T );Y )) is dense in L2(W) and consequently in every L2,Q(W)and every (S)ρ,γ, −1 < ρ ≤ 1, γ ∈ R.

Definition 7.9. If u ∈ L2,Q(W; X) for some Q, or if u ∈ ⋃q≥0(S)−ρ,−γ(X), 0 ≤ ρ ≤ 1,

then the deterministic function

(7.12) Su(h) =∑

α∈J

uαhα

√α!

∈ X

is called the S-transform of u. Similarly, for g ∈ D′ (Y ; L2,Q(W; X)) the S-transformSg(h) ∈ D′(Y ; X) is defined by setting (Sg(h))k = (Sgk)(h).

Note that if u ∈ L2(W; X), then Su(h) = E(uE(h)). If u belongs to L2,Q(W; X) or to⋃q≥0(S)−ρ,−γ(X), 0 ≤ ρ < 1, then Su(h) is defined for all h ∈ D (L2((0, T );Y )) . If u ∈⋃γ≥0(S)−1,−γ(X), then Su(h) is defined only for h sufficiently close to zero.

By Remark 7.8, an element u from L2,Q(W; X) or⋃

γ≥0(S)−ρ,−γ(X), 0 ≤ ρ < 1, is uniquelydetermined by the collection of deterministic functions Su(h), h ∈ D (L2((0, T );Y )) . SinceE(h) > 0 for all h ∈ D (L2((0, T );Y )), Remark 7.8 also suggests the following definition.

Definition 7.10. An element u from L2,Q(W) or⋃

γ≥0(S)−ρ,−γ, 0 ≤ ρ < 1 is called non-negative (u ≥ 0) if and only if Su(h) ≥ 0 for all h ∈ D (L2((0, T );Y )).

The definition of the operator Q and Definition 7.10 imply the following result.


Proposition 7.11. A generalized random element u from L2,Q(W) is non-negative if andonly if Qu ≥ 0.

For example, the solution of equation (7.7) is non-negative because

Qu(t) = exp

∑

k≥1

(qkwk(t)− (1/2)q2k)

.

We conclude this section with one technical remark.

Definition 7.9 expresses the S-transform in terms of the generalized Fourier coefficients.The following results makes it possible to recover generalized Fourier coefficients from thecorresponding S-transform.

Proposition 7.12. If u belongs to some L2,Q(W; X) or⋃

γ≥0(S)−ρ,−γ(X), 0 ≤ ρ ≤ 1, then

(7.13) uα =1√α!

∏

i,k

∂αki Su(h)

∂hαk

ik,i

∣∣∣∣∣∣h=0

.

Proof. For each α ∈ J with K non-zero entries, equality (7.12) and Lemma 7.7 imply thatthe function Su(h), as a function of K variables hk,i, is analytic in some neighborhood ofzero. Then (7.13) follows after differentiation of the series (7.12). ¤

8. General Properties of the Wiener Chaos Solutions

Using notations and assumptions from Section 6, consider the linear evolution equation

(8.1) du(t) = (Au(t) + f(t))dt + (Mu(t) + g(t), dW (t))Y , 0 < t ≤ T, u|t=0 = u0.

The objective of this section is to study how the Wiener Chaos compares with the traditionaland white noise solutions.

To make the presentation shorter, call an X-valued generalized random element S-admissibleif and only if it belongs to L2,Q(FW ;X) for some Q or to (S)ρ,q(X) for some ρ ∈ [−1, 1]and q ∈ R. It was shown in Section 7 that, for every S-admissible u, the S-transform Su(h)is defined when h =

∑i,k hk,imiyk ∈ D(L2((0, T );Y )) and is an analytic function of hk,i in

some neighborhood of h = 0.

The next result describes the S-transform of the Wiener Chaos solution.

Theorem 8.1. Assume that

(1) there exists a unique w(A,X) Wiener Chaos solution u of (8.1) and u is S-admissible;

(2) For each t ∈ [0, T ], the linear operators A(t),Mk(t) are bounded from A to X;(3) the generalized random elements u0, f, gk are S-admissible.

Then, for every h ∈ D(L2((0, T );Y )) with ‖h‖2L2((0,T );Y ) sufficiently small, the function

v = Su(h) is a w(A,X) solution of the deterministic equation

(8.2) v(t) = Su0(h) +∫ t

0

(Av + Sf(h) + (Mkv + Sgk(h))hk

)(s)ds.


Proof. By assumption, Su(h) exists for suitable functions h. Then the S-transformed equa-tion (8.2) follows from the definition of the S-transform (7.12) and the propagator equation(6.4) satisfied by the generalized Fourier coefficients of u. Indeed, continuity of operator Aimplies

S(Au)(h) =∑α

hα

√α!Auα = A

∑α

hα

√α!

uα = A(Su(h)).

Similarly,

∑α

hα

√α!

∑

i,k

√αk

iMkuα−(i,k)mi =∑α

∑

i,k

hα−(i,k)

√α−(i, k)!

Mkuα−(i,k)mihk,i

=∑

i,k

(∑α

hα

√αMkuα

)mihk,i = Mk(Su(h))hk.

Computations for the other terms are similar. Theorem 8.1 is proved. ¤

Remark 8.2. If h ∈ D(L2((0, T );Y )) and

(8.3) Et(h) = exp(∫ t

0(h(s), dW (s))Y − 1

2

∫ t

0‖h(t)‖2

Y dt

),

then, by the Ito formula,

(8.4) dEt(h) = Et(h)(h(t), dW (t))Y .

If u0 is deterministic, f and gk are FWt -adapted, and u is a square-integrable solution of

(8.1), then equality (8.2) is obtained by multiplying equations (8.4) and (8.1) according tothe Ito formula and taking the expectation.

Remark 8.3. Rewriting (8.4) as

dEt(h) = Et(h)hk,imi(t)dwk(t)

and using the relations

Et(h) = E(ET (h)|FWt ), ξα =

1√α!

∏

i,k

∂αki ET (h)

∂hαk

ik,i

∣∣∣∣∣∣h=0

,

we arrive at representation (5.5) for E(ξα|FWt ).

A partial converse of Theorem 8.1 is that, under some regularity conditions, the WienerChaos solution can be recovered from the solution of the S-transformed equation (8.2).

Theorem 8.4. Assume that the linear operators A(t), Mk(t), t ∈ [0, T ], are bounded fromA to X, the input data u0, f , gk are S-admissible, and, for every h ∈ D(L2((0, T );Y ))with ‖h‖2

L2((0,T );Y ) sufficiently small, there exists a w(A,X) solution v = v(t;h) of equation(8.2). We write h = hk,imiyk and consider v as a function of the variables hk,i. Assumethat all the derivatives of v at the point h = 0 exists, and, for α ∈ J , define

(8.5) uα(t) =1√α!

∏

i,k

∂αki v(t; h)

∂hαk

ik,i

∣∣∣∣∣∣h=0

.

Then the generalized random process u(t) =∑

α∈J uα(t)ξα is a w(A,X) Wiener Chaossolution of (8.1).


Proof. Differentiation of (8.2) and application of Proposition 7.12 show that the functionsuα satisfy the propagator (6.4). ¤

Remark 8.5. The central part in the construction of the white noise solution of (8.1)is proving that the solution of (8.2) is an S-transform of a suitable generalized randomprocess. For many particular cases of equation (8.1), the corresponding analysis is carriedout in [10, 12, 33, 40]. The consequence of Theorems 8.1 and 8.4 is that a white noisesolution of (8.1), if exists, must coincide with the Wiener Chaos solution.

The next theorem establishes the connection between the Wiener Chaos solution and thetraditional solution. Recall that the traditional, or square-integrable, solution of (8.1) wasintroduced in Definition 2.2. Accordingly, the notations from Section 2 will be used.

Theorem 8.6. Let (V,H, V ′) be a normal triple of Hilbert spaces. Take a deterministicfunction u0 and FW

t -adapted random processes function, f and gk so that (2.3) holds. Underthese assumptions we have the following two statements.

(1) An FWt -adapted traditional solution of (8.1) is also a Wiener Chaos solution.

(2) If u is a w(V, V ′) Wiener Chaos solution of (8.1) so that

(8.6)∑

α∈J

(∫ T

0‖uα(t)‖2

V dt + sup0≤t≤T

‖uα(t)‖2H

)< ∞,

then u is an FWt -adapted traditional solution of (8.1).

Proof. (1) If u = u(t) is an FWt -adapted traditional solution, then

uα(t) = E(u(t)ξα) = E(u(t)E(ξα|FW

t ))

= E(u(t)ξα(t)).

Then the propagator (6.4) for uα follows after applying the Ito formula to the productu(t)ξα(t) and using (5.5).

(2) Assumption (8.6) implies

u ∈ L2(Ω× (0, T );V )⋂

L2(Ω;C((0, T );H)).

Then, by Theorem 8.1, for every ϕ ∈ V and h ∈ D((0, T );Y ), the S-transform uh of usatisfies

(uh(t), ϕ)H = (u0, ϕ)H +∫ t

0〈Auh(s), ϕ〉ds +

∫ t

0〈f(s), ϕ〉ds

+∑

α∈J

hα

α!

∑

i,k

∫ t

0

√αk

i mi(s)((Mkuα−(i,k)(s), ϕ)H

+ (gk(s), ϕ)HI(|α| = 1))ds.

If I(t) =∫ t0 (Mku(s), ϕ)Hdwk(s), then

(8.7) E(I(t)ξα(t)) =∫ t

0

∑

i,k

√αk

i mi(s)(Mkuα−(i,k)(s), ϕ)Hds.

Similarly,

E(

ξα(t)∫ t

0(gk(s), ϕ)Hdwk(s)

)=

∑

i,k

∫ t

0

√αk

i mi(s)(gk(s), ϕ)HI(|α| = 1)ds.


Therefore,∑

α∈J

hα

α!

∑

i,k

∫ t

0

√αk

i mi(s)(Mkuα−(i,k)(s), ϕ)Hds

= E(E(h)

∫ t

0((Mku(s), ϕ)H + (gk(s), ϕ)H) dwk(s)

).

As a result,

E (E(h)(u(t), ϕ)H) = E (E(h)(u0, ϕ)H)

+ E(E(h)

∫ t

0〈Au(s), ϕ〉ds

)+ E

(E(h)

∫ t

0〈f(s), ϕ〉ds

)

+ E(E(h)

∫ t

0((Mku(s), ϕ)H + (gk(s), ϕ)H) dwk(s)

).

(8.8)

Equality (8.8) and Remark 7.8 imply that, for each t and each ϕ, (2.4) holds with probabilityone. Continuity of u implies that, for each ϕ, a single probability-one set can be chosen forall t ∈ [0, T ]. Theorem 9.6 is proved. ¤

9. Regularity of the Wiener Chaos Solution

Let F = (Ω,F , Ftt≥0,P) be a stochastic basis with the usual assumptions and wk =wk(t), k ≥ 1, t ≥ 0, a collection of standard Wiener processes on F. As in Section 2, let(V, H, V ′) be a normal triple of Hilbert spaces and A(t) : V → V ′, Mk(t) : V → H, linearbounded operators; t ∈ [0, T ].

In this section we study the linear equation

(9.1) u(t) = u0 +∫ t

0(Au(s) + f(s))ds +

∫ t

0(Mku(s) + gk(s))dwk, 0 ≤ t ≤ T,

under the following assumptions:

A1 There exist positive numbers C1 and δ so that

(9.2) 〈A(t)v, v〉+ δ‖v‖2V ≤ C1‖v‖2

H , v ∈ V, t ∈ [0, T ].

A2 There exists a real number C2 so that

(9.3) 2〈A(t)v, v〉+∑

k≥1

‖Mk(t)v‖2H ≤ C2‖v‖2

H , v ∈ V, t ∈ [0, T ].

A3 The initial condition u0 is non-random and belongs to H; the process f = f(t) isdeterministic and

∫ T0 ‖f(t)‖2

V ′dt < ∞; each gk = gk(t) is a deterministic processesand

∑k≥1

∫ T0 ‖gk(t)‖2

Hdt < ∞.

Note that condition (9.3) is weaker than (2.5). Traditional analysis of equation (9.1) under(9.3) requires additional regularity assumptions on the input data and additional Hilbertspace constructions beyond the normal triple [42, Section 3.2]. In particular, no existence ofa traditional solution is known under assumptions A1-A3, and the Wiener chaos approachprovides new existence and regularity results for equation (9.1). A different version of thefollowing theorem is presented in [29].

Theorem 9.1. Under assumptions A1–A3, for every T > 0, equation (9.1) has a uniquew(V, V ′) Wiener Chaos solution. This solution u = u(t) has the following properties:


(1) There exists a weight sequence Q so that

u ∈ L2,Q(W; L2((0, T );V ))⋂

L2,Q(W;C((0, T );H)).

(2) For every 0 ≤ t ≤ T , u(t) ∈ L2(Ω; H) and

(9.4) E‖u(t)‖2H ≤ 3eC2t

‖u0‖2

H + Cf

∫ t

0‖f(s)‖2

V ′ds +∑

k≥1

∫ t

0‖gk(s)‖2

Hds

,

where the number C2 is from (9.3) and the positive number Cf depends only on δand C1 from (9.2).

(3) For every 0 ≤ t ≤ T ,

u(t) = u(0) +∑

n≥1

∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0

Φt,snMkn · · ·Φs2,s1

(Mk1u(0) + gk1(s1))dwk1(s1) · · · dwkn(sn),

(9.5)

where Φt,s is the semi-group of the operator A.

Proof. Assumption A2 and the properties of the normal triple imply that there exists apositive number C∗ so that

(9.6)∑

k≥1

‖Mk(t)v‖2H ≤ C∗‖v‖2

V , v ∈ V, t ∈ [0, T ].

Define the sequence Q so that

(9.7) qk =(

µδ

C∗

)1/2

:= q, k ≥ 1,

where µ ∈ (0, 2) and δ is from Assumption A1. Then, by Assumption A2,

(9.8) 2〈Av, v〉+∑

k≥1

q2‖Mkv‖2H ≤ −(2− µ)δ‖v‖2

V + C1‖v‖2H .

It follows from Theorem 2.4 that equation

(9.9) v(t) = u0 +∫ t

0(Av + f)(s)ds +

∑

k≥1

∫ t

0q(Mkv + gk)(s)dwk(s)

has a unique solution

v ∈ L2(W;L2((0, T );V ))⋂

L2(W;C((0, T );H)).

Comparison of the propagators for equations (9.1) and (9.9) shows that u = Q−1v is theunique w(V, V ′) solution of (9.1) and

(9.10) u ∈ L2,Q(W; L2((0, T );V ))⋂

L2,Q(W;C((0, T );H)).

If C∗ < 2δ, then equation (9.1) is strongly parabolic and q > 1 is an admissible choice ofthe weight. As a result, for strongly parabolic equations, the result (9.10) is stronger thanthe conclusion of Theorem 2.4.


The proof of (9.4) is based on the analysis of the propagator

uα(t) = u0I(|α| = 0) +∫ t

0

(Auα(s) + f(s)I(|α| = 0)

)ds

+∫ t

0

∑

i,k

√αk

i (Mkuα−(i,k)(s) + gk(s)I(|α| = 1))mi(s)ds.(9.11)

We consider three particular cases: (1) f = gk = 0 (the homogeneous equation); (2)u0 = gk = 0; (3) u0 = f = 0. The general case will then follow by linearity and the triangleinequality.

Denote by (Φt,s, t ≥ s ≥ 0) the semi-group generated by the operator A(t); Φt := Φt,0. Oneof the consequence of Theorem 2.4 is that, under Assumption A1, this semi-group existsand is strongly continuous in H.

Consider the homogeneous equation: f = gk = 0. By Corollary 6.6,

(9.12)∑

|α|=n

‖uα(t)‖2H =

∑

k1,...,kn≥1

∫ t

0

∫ sn

0· · ·

∫ s2

0‖Φt,snMkn · · ·Φs2,s1Mk1Φs1u0‖2

Hdsn,

where dsn = ds1 . . . dsn. Define Fn(t) =∑|α|=n ‖uα(t)‖2

H , n ≥ 0. Direct application of(9.3) shows that

(9.13)d

dtF0(t) ≤ C2F0(t)−

∑

k≥1

‖MkΦtu0‖2H .

For n ≥ 1, equality (9.12) implies

d

dtFn(t) =

∑

k1,...,kn≥1

∫ t

0

∫ sn−1

0· · ·

∫ s2

0‖MknΦt,sn−1 · · ·Mk1Φs1u0‖2

Hdsn−1

+∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0〈AΦt,snMkn . . .Φs1u0,Φt,snMkn . . .Φs1u0〉dsn.

(9.14)

By (9.3),∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0〈AΦt,snMkn . . .Φs1u0, Φt,snMkn . . .Φs1u0〉dsn

≤ −∑

k1,...,kn+1≥1

∫ t

0

∫ sn

0. . .

∫ s2

0‖Mkn+1Φt,snMkn . . .Mk1Φs1u0‖2

Hdsn

+C2

∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0‖Φt,snMkn . . .Mk1Φs1u0‖2

Hdsn.

(9.15)

As a result, for n ≥ 1,d

dtFn(t) ≤ C2Fn(t)

+∑

k1,...,kn≥1

∫ t

0

∫ sn−1

0. . .

∫ s2

0‖MknΦt,sn−1Mkn−1 . . .Mk1Φs1u0‖2

Hdsn−1

−∑

k1,...,kn+1≥1

∫ t

0

∫ sn

0. . .

∫ s2


Hdsn.

(9.16)


Consequently,

(9.17)d

dt

N∑

n=0

∑

|α|=n

‖uα(t)‖2H ≤ C2

N∑

n=0

∑

|α|=n

‖uα(t)‖2H ,

so that, by the Gronwall inequality,

(9.18)N∑

n=0

∑

|α|=n

‖uα(t)‖2H ≤ eC2t‖u0‖2

H

or

(9.19) E‖u(t)‖2H ≤ eC2t‖u0‖2

H .

Next, let us assume that u0 = gk = 0. Then the propagator (9.11) becomes

(9.20) uα(t) =∫ t

0(Auα(s) + f(s)I(|α| = 0))ds +

∫ t

0

∑

i,k

√αk

iMkuα−(i,k)(s)mi(s)ds.

Denote by u(0)(t) the solution corresponding to α = 0. Note that

‖u(0)(t)‖2H = 2

∫ t

0〈Au(0)(s), u(0)(s)〉ds + 2

∫ t

0〈f(s), u(0)(s)〉ds

≤ C2

∫ t

0‖u(0)(s)‖2

Hds−∫ t

0

∑

k≥1

‖Mku(0)(s)‖2Hds + Cf

∫ t

0‖f(s)‖2

V ′ds.

By Corollary 6.6,

(9.21)∑

|α|=n

‖uα(t)‖2H =

∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0‖Φt,snMkn . . .Mk1u(0)(s1)‖2

Hdsn

for n ≥ 1. Then, repeating the calculations (9.14)–(9.16), we conclude that

(9.22)N∑

n=1

∑

|α|=n

‖uα(t)‖2H ≤ Cf

∫ t

0‖f(s)‖2

V ′ds + C2

∫ t

0

N∑

n=1

∑

|α|=n

‖uα(s)‖2Hds,

and, by the Gronwal inequality,

(9.23) E‖u(t)‖2H ≤ CfeC2t

∫ t

0‖f(s)‖2

V ′ds.

Finally, let us assume that u0 = f = 0. Then the propagator (9.11) becomes

uα(t) =∫ t

0Auα(s)ds

+∫ t

0

∑

i,k

√αk

iMkuα−(i,k)(s) + gk(s)I(|α| = 1)

mi(s)ds.

(9.24)

Even though uα(t) = 0 if α = 0, we have

(9.25) u(ik) =∫ t

0Φt,sgk(s)mi(s)ds,


and then the arguments from the proof of Corollary 6.6 apply, resulting in∑

|α|=n

‖uα(t)‖2H =

∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0‖Φt,snMkn . . . Φs2,s1gk1(s1)‖2

Hdsn

for n ≥ 1. Note that∑

|α|=1

‖uα(t)‖2H =

∑

k≥1

∫ t

0‖gk(s)‖2

Hds + 2∑

k≥1

∫ t

0〈AΦt,sgk(s), Φt,sgk(s)〉ds.

Then, repeating the calculations (9.14)–(9.16), we conclude that

(9.26)N∑

n=1

∑

|α|=n

‖uα(t)‖2H ≤

∑

k≥1

∫ t

0‖gk(s)‖2

Hds + C2

∫ t

0

N∑

n=1

∑

|α|=n

‖uα(s)‖2Hds,

and, by the Gronwal inequality,

(9.27) E‖u(t)‖2H ≤ eC2t

∑

k≥1

∫ t

0‖gk(s)‖2

Hds.

To derive (9.4), it remains to combine (9.19), (9.23), and (9.27) with the inequality (a+ b+c)2 ≤ 3(a2 + b2 + c2).

Representation (9.5) of the Wiener chaos solution as a sum of iterated Ito integrals nowfollows from Corollary 6.6. Theorem 9.1 is proved. ¤

Corollary 9.2. If∑

α∈J

∫ T

0‖uα(s)‖2

V ds < ∞, then∑

α∈Jsup

0≤t≤T‖uα(t)‖2

H < ∞.

Proof. The proof of Theorem 9.1 shows that it is enough to consider the homogeneousequation. Then by inequalities (9.15)–(9.16),

n1∑

`=n+1

∑

|α|=`

‖uα(t)‖2H =

n1∑

`=n+1

F`(t)

≤ eC2T∑

k1,...,kn+1≥1

∫ T

0

∫ t

0

∫ sn

0. . .

∫ s2

0‖Mkn+1Φt,snMkn . . .Φs1u0‖2

Hdsndt.

(9.28)

By Corollary 6.6,∫ T

0‖uα(s)‖2

V ds

=∑

n≥1

∑

k1,...,kn≥1

∫ T

0

∫ t

0

∫ sn

0. . .

∫ s2

0‖MknΦt,snMkn . . .Φs1u0‖2

V dsndt < ∞.(9.29)

As a result, (9.6) and (9.29) imply

limn→∞

∫ T

0

∫ t

0

∫ sn

0. . .

∫ s2


Hdsndt = 0,

which, by (9.28), implies uniform, with respect to t, convergence of the series∑

α∈J ‖uα(t)‖2H .

Corollary 9.2 is proved. ¤


Corollary 9.3. Let aij , bi, c, σik, νk be deterministic measurable functions of (t, x) so that

|aij(t, x)|+ |bi(t, x)|+ |c(t, x)|+ |σik(t, x)|+ |νk(t, x)| ≤ K,

i, j = 1, . . . , d, k ≥ 1, x ∈ Rd, 0 ≤ t ≤ T ;(aij(t, x)− 1

2σik(t, x)σjk(t, x)

)yiyj ≥ 0,

x, y ∈ Rd, 0 ≤ t ≤ T ; and ∑

k≥1

|νk(t, x)|2 ≤ Cν < ∞,

x ∈ Rd, 0 ≤ t ≤ T. Consider the equation

(9.30) du = (Di(aijDju) + biDiu + c u + f)dt + (σikDiu + νku + gk)dwk.

Assume that the input data satisfy u0 ∈ L2(Rd), f ∈ L2((0, T );H−12 (Rd)),∑

k≥1 ‖gk‖2L2((0,T )×Rd)

< ∞, and there exists an ε > 0 so that

aij(t, x)yiyj ≥ ε|y|2, x, y ∈ Rd, 0 ≤ t ≤ T.

Then there exists a unique Wiener Chaos solution u = u(t, x) of (9.30). The solution hasthe following regularity:

(9.31) u(t, ·) ∈ L2(W; L2(Rd)), 0 ≤ t ≤ T,

and

E‖u‖2L2(Rd)(t) ≤ C∗

(‖u0‖2

L2(Rd) + ‖f‖2L2((0,T );H−1

2 (Rd))

+∑

k≥1

‖gk‖2L2((0,T )×Rd)

),

(9.32)

where the positive number C∗ depends only on Cν ,K, T, and ε.

Remark 9.4.(1) If (2.5) holds instead of (9.3), then the proof of Theorem 9.1, in particular, (9.15)–(9.16), shows that the term E‖u(t)‖2

H in the left-hand-side of inequality (9.4) can be replacedwith

E(‖u(t)‖2

H + ε

∫ t

0‖u(s)‖2

V ds

).

(2) If f = gk = 0 and the equation is fully degenerate, that is, 2〈A(t)v, v〉+∑k≥1 ‖Mk(t)v‖2

H =0, t ∈ [0, T ], then it is natural to expect conservation of energy. Once again, analysis of(9.15)–(9.16) shows that equality

E‖u(t)‖2H = ‖u0‖2

H

holds if and only if

limn→∞

∫ T

0

∫ t

0

∫ sn

0. . .

∫ s2


Hdsndt = 0.

The proof of Corollary 9.2 shows that a sufficient condition for the conservation of energyin a fully degenerate homogeneous equation is E

∫ T0 ‖u(t)‖2

V dt < ∞.

One of applications of the Wiener Chaos solution is new numerical methods for solving theevolution equations. Indeed, an approximation of the solution is obtained by truncating thesum

∑α∈J uα(t)ξα. For the Zakai filtering equation, these numerical methods were studied

in [25, 26, 27]; see also Section 11 below. The main question in the analysis is the rate of


convergence, in n, of the series∑

n≥1

∑|α|=n ‖u(t)‖2

H . In general, this convergence can bearbitrarily slow. For example, consider the equation

du =12uxxdt + uxdw(t), t > 0, x ∈ R,

in the normal triple (H12 (R), L2(R),H−1

2 (R)), with initial condition u|t=0 = u0 ∈ L2(R). Itfollows from (9.12) that

Fn(t) =∑

|α|=n

‖u‖2L2(R)(t) =

tn

n!

∫

R|y|2ne−y2t|u0|2dy,

where u0 is the Fourier transform of u0. If

|u0(y)|2 =1

(1 + |y|2)γ, γ > 1/2,

then the rate of decay of Fn(t) is close to n−(1+2γ)/2. Note that, in this example,E‖u‖2

L2(R)(t) = ‖u0‖2L2(R).

An exponential convergence rate that is uniform in ‖u0‖2H is achieved under strong parabol-

icity condition (2.5). An even faster factorial rate is achieved when the operators Mk arebounded on H.

Theorem 9.5. Assume that the there exist a positive number ε and a real number C0 sothat

2〈A(t)v, v〉+∑

k≥1

‖Mk(t)v‖2H + ε‖v‖2

V ≤ C0‖v‖2H , t ∈ [0, T ], v ∈ V.

Then there exists a positive number b so that, for all t ∈ [0, T ],

(9.33)∑

|α|=n

‖uα(t)‖2H ≤ ‖u0‖2

H

(1 + b)n.

If, in addition,∑

k≥1 ‖Mk(t)ϕ‖2H ≤ C3‖ϕ‖2

H , then

(9.34)∑

|α|=n

‖uα(t)‖2H ≤ (C3t)n

n!eC1t‖u0‖2

H .

Proof. If C∗ is from (9.6) and b = ε/C∗, then the operators√

1 + bMk satisfy

2〈A(t)v, v〉+ (1 + b)∑

k≥1

‖Mk(t)‖2H ≤ C0‖v‖2

H .

By Theorem 9.1,

(1 + b)n∑

k1,...,kn≥1

∫ t

0

∫ sn

0. . .

∫ s2

0‖Φt,snMkn . . .Mk1Φs1u0‖2

Hdsn ≤ ‖u0‖2H ,

and (9.33) follows.

To establish (9.34), note that, by (9.2),

‖Φtf‖2H ≤ eC1t‖f‖2

H ,

and therefore the result follows from (9.12). Theorem 9.5 is proved. ¤

The Wiener Chaos solution of (9.1) is not, in general, a solution of the equation in thesense of Definition 2.2. Indeed, if u 6∈ L2(Ω × (0, T );V ), then the expressions 〈Au(s), ϕ〉


and (Mku(s), ϕ)H are not defined. On the other hand, if there is a possibility to move theoperators A and M from the solution process u to the test function ϕ, then equation (9.1)admits a natural analog of the traditional weak formulation (2.4).

Theorem 9.6. In addition to A1–A3, assume that there exist operators A∗(t), M∗k(t) and

a dense subset V0 of the space V so that

(1) A∗(t)(V0) ⊆ H, M∗k(t)(V0) ⊆ H, t ∈ [0, T ];

(2) for every v ∈ V , ϕ ∈ V0, and t ∈ [0, T ], 〈A(t)v, ϕ〉 = (v,A∗(t)ϕ)H , (Mk(t)v, ϕ)H =(v,M∗

k(t)ϕ)H .

If u = u(t) is the Wiener Chaos solution of (9.1), then, for every ϕ ∈ V0 and every t ∈ [0, T ],the equality

(u(t), ϕ)H = (u0, ϕ)H +∫ t

0(u(s),A∗(s)ϕ)Hds +

∫ t

0〈f(s), ϕ〉ds

+∫ t

0(u(s),M∗

k(s)ϕ)Hdwk(s) +∫ t

0(gk(s), ϕ)Hdwk(s)

(9.35)

holds in L2(W).

Proof. The arguments are identical to the proof of Theorem 8.6(2). ¤

As was mentioned earlier, the Wiener Chaos solution can be constructed for anticipatingequations, that is, equations with FW

T -measurable input data. With obvious modifications,inequality (9.4) holds if each of the input functions u0, f , and gk in (9.1) is a finite linearcombination of the basis elements ξα. The following example demonstrates that inequality(9.4) is impossible for general anticipating equation.

Example 9.7. Let u = u(t, x) be a Wiener Chaos solution of an ordinary differentialequation

(9.36) du = udw(t), 0 < t ≤ 1,

with u0 =∑

α∈J aαξα. For n ≥ 0, denote by (n) the multi-index with α1 = n and αi = 0,i ≥ 2, and assume that a(n) > 0, n ≥ 0. Then

(9.37) Eu2(1) ≥ C∑

n≥0

e√

na2(n).

Indeed, the first column of propagator for α = (n) is u(0)(t) = a(0) and

u(n)(t) = a(n) +√

n

∫ t

0u(n−1)(s)ds,

so that

u(n)(t) =n∑

k=0

√n!√

(n− k)!k!

a(n−k)√k!

tk.

Then u2(n)(1) ≥ ∑n

k=0

(nk

)a2(n−k)

k! and

∑

n≥0

u2(n)(1) ≥

∑

n≥0

∑

k≥0

1k!

(n + k

n

) a2

(n).


Since∑

k≥0

1k!

(n + k

n

)≥

∑

k≥0

nk

(k!)2≥ Ce

√n,

the result follows.

The consequence of Example 9.7 is that it is possible, in (9.1), to have u0 ∈ Ln2 (W;H) for

every n, and still get E‖u(t)‖2H = +∞ for all t > 0. More generally, the solution operator

for (9.1) is not bounded on any L2,Q or (S)−ρ,−γ . On the other hand, the following resultholds.

Theorem 9.8. In addition to Assumptions A1, A2, let u0 be an element of D′(W; H),f , an element of D′(W; L2((0, T ), V ′)), and each gk, an element of D′(W; L2((0, T ),H)).Then the Wiener Chaos solution of equation (9.1) satisfies

√√√√∑

α∈J

‖uα(t)‖2H

α!≤ C

∑

α∈J

1√α!

(‖u0α‖H +

(∫ t

0‖fα(s)‖2

V ′ds

)1/2

+

∑

k≥1

∫ t

0‖gk,α(s)‖2

Hds

1/2 ),

(9.38)

where C > 0 depends only on T and the numbers δ, C1, and C2 from (9.2) and (9.3).

Proof. To simplify the presentation, assume that f = gk = 0. For fixed γ ∈ J , denote byu(t; ϕ; γ) the Wiener Chaos solution of the equation (9.1) with initial condition u(0;ϕ; γ) =ϕξγ . Denote by (0) the zero multi-index. The structure of the propagator implies thefollowing relation:

(9.39)uα+γ(t; ϕ; γ)√

(α + γ)!=

uα

(t; ϕ√

γ!; (0)

)√

α!.

Clearly, uα(t; ϕ; γ) = 0 if |α| < |γ|. If

‖v(t)‖2(S)−1,0(H) =

∑

α∈J

‖vα(t)‖2H

α!,

then, by linearity and triangle inequality,

‖u(t)‖(S)−1,0(H) ≤∑

γ∈J‖u(t; u0γ ; γ)‖(S)−1,0(H).

We also have by (9.39) and Theorem 9.1

‖u(t; u0γ ; γ)‖2(S)−1,0(H) =

∥∥∥∥u

(t;

u0γ√γ!

; (0))∥∥∥∥

2

(S)−1,0(H)

≤ E∥∥∥∥u

(t;

u0γ√γ!

; (0))∥∥∥∥

2

H

≤ eC2t ‖u0γ‖2H

γ!.

Inequality (9.38) then follows. Theorem 9.8 is proved. ¤


Remark 9.9. Using Proposition 7.2 and the Cauchy-Schwartz inequality, (9.38) can bere-written in a slightly weaker form to reveal continuity of the solution operator for equation(9.1) from (S)−1,γ to (S)−1,0 for every γ > 1:

‖u(t)‖2(S)−1,0(H) ≤ C

(‖u0‖2

(S)−1,γ(H) +∫ t

0‖f(s)‖2

(S)−1,γ(V ′)ds

+∑

k≥1

∫ t

0‖gk(s)‖2

(S)−1,γ(H)ds

).

10. Probabilistic Representation of Wiener Chaos Solutions

The general discussion so far has been dealing with the abstract evolution equation

du = (Au + f)dt +∑

k≥1

(Mku + gk)dwk.

By further specifying the operators A and Mk, as well as the input data u0, f, and gk, it ispossible to get additional information about the Wiener Chaos solution of the equation.

Definition 10.1. For r ∈ R, the space L2,(r) = L2,(r)(Rd) is the collection of real-valuedmeasurable functions so that f ∈ L2,(r) if and only if

∫Rd |f(x)|2(1+|x|2)rdx < ∞. The space

H12,(r) = H1

2,(r)(Rd) is the collection of real-valued measurable functions so that f ∈ H1

2,(r) ifand only if f and all the first-order generalized derivatives Dif of f belong to L2,(r).

It is known, for example, from Theorem 3.4.7 in [42], that L2,(r) is a Hilbert space withnorm

‖f‖20,(r) =

∫

Rd

|f(x)|2(1 + |x|2)rdx,

and H12,(r) is a Hilbert space with norm

‖f‖1,(r) = ‖f‖0,(r) +d∑

i=1

‖Dif‖0,(r).

Denote by H−12,(r) the dual of H1

2,(r) with respect to the inner product in L2,(r). Then

(H12,(r), L2,(r),H

−12,(r)) is a normal triple of Hilbert spaces.

Let F = (Ω,F , Ftt≥0,P) be a stochastic basis with the usual assumptions and wk =wk(t), k ≥ 1, t ≥ 0, a collection of standard Wiener processes on F. Consider the linearequation

(10.1) du = (aijDiDju + biDiu + cu + f)dt + (σikDiu + νku + gk)dwk

under the following assumptions:

B0 All coefficients, free terms, and the initial condition are non-random.B1 The functions aij = aij(t, x) and their first-order derivatives with respect to x are

uniformly bounded in (t, x), and the matrix (aij) is uniformly positive definite, thatis, there exists a δ > 0 so that, for all vectors y ∈ Rd and all (t, x), aijyiyj ≥ δ|y|2.

B2 The functions bi = bi(t, x), c = c(t, x), and νk = νk(t, x) are measurable andbounded in (t, x).

B3 The functions σik = σik(t, x) are continuous and bounded in (t, x).


B4 The functions f = f(t, x) and gk = gk(t, x) belong to L2((0, T );L2,(r)) for somer ∈ R.

B5 The initial condition u0 = u0(x) belongs to L2,(r).

Under Assumptions B2–B4, there exists a sequence Q = qk, k ≥ 1 of positive numberswith the following properties:

P1 The matrix A with Aij = aij − (1/2)∑

k≥1 qkσikσjk satisfies

Aij(t, x)yiyj ≥ 0,

x, y ∈ Rd, 0 ≤ t ≤ T .P2 There exists a number C > 0 so that

∑

k≥1

(supt,x|qkνk(t, x)|2 +

∫ T

0‖qkgk‖p

0,(r)(t)dt

)≤ C.

For the matrix A and each t, x, we have Aij(t, x) = σik(t, x)σjk(t, x), where the functionsσik are bounded. This representation might not be unique; see, for example, [7, TheoremIII.2.2] or [44, Lemma 5.2.1]. Given any such representation of A, consider the followingbackward Ito equation

Xt,x,i (s) = xi +∫ t

sBi (τ, Xt,x (τ)) dτ +

∑

k≥1

qkσik (τ, Xt,x (τ))←−−dwk (τ)

+∫ t

sσik (τ,Xt,x (τ))

←−dwk (τ) ; s ∈ (0, t), t ∈ (0, T ], t− fixed,

(10.2)

where Bi = bi −∑

k≥1 q2kσikνk and wk, k ≥ 1, are independent standard Wiener processes

on F that are independent of wk, k ≥ 1. This equation might not have a strong solution,but does have weak, or martingale, solutions due to Assumptions B1–B3 and propertiesP1 and P2 of the sequence Q; this weak solution is unique in the sense of probability law[44, Theorem 7.2.1].

The following result is a variation of Theorem 4.1 in [29].

Theorem 10.2. Under assumptions B0–B5 equation (10.1) has a uniquew(H1

2,(r),H−12,(r)) Wiener Chaos solution. If Q is a sequence with properties P1 and P2,

then the solution of (10.1) belongs to

L2,Q

(W; L2((0, T );H1

2,(r)))⋂

L2,Q

(W;C((0, T );L2,(r))

)

and has the following representation:

u(t, x) = Q−1E

(∫ t

0f(s,Xt,x(s))γ(t, s, x)ds

+∑

k≥1

∫ t

0qkgk(s,Xt,x(s))γ(t, s, x)

←−−dwk(s) + u0(Xt,x(0))γ(t, 0, x)

∣∣∣FWt

), t ≤ T,

(10.3)


where Xt,x(s) is a weak solution of (10.2), and

γ(t, s, x) = exp

(∫ t

sc(τ,Xt,x(τ))dτ +

∑

k≥1

∫ t

sqkνk(τ, Xt,x(τ))

←−−dwk(τ)

− 12

∫ t

s

∑

k≥1

q2k|νk(τ, Xt,x(τ))|2dτ

).

(10.4)

Proof. It is enough to establish (10.3) when t = T . Consider the equation

(10.5) dU = (aijDiDjU + biDiU + cU + f)dt +∑

k≥1

(σikDiU + νkU + gk)qkdwk

with initial condition U(0, x) = u0(x). Applying Theorem 2.4 in the normal triple(H1

2,(r), L2,(r),H−12,(r)), we conclude that there is a unique solution

U ∈ L2

(W; L2((0, T );H1

2,(r))) ⋂

L2

(W;C((0, T );L2,(r))

)

of this equation. By Proposition 7.4, the process u = Q−1U is the corresponding WienerChaos solution of (10.1). To establish representation (10.3), consider the S-transform Uh

of U . According to Theorem 8.1, the function Uh is the unique w(H12,(r),H

−12,(r)) solution of

the equation

(10.6) dUh = (aijDiDjUh + biDiUh + cUh + f)dt +∑

k≥1

(σikDiUh + νkUh + gk)qkhkdt

with initial condition Uh|t=0 = u0. We also define

Y (T, x) =∫ T

0f(s, XT,x(s))γ(T, s, x)ds

+∑

k≥1

∫ T

0gk(s,XT,x(s))γ(T, s)qk

←−−dwk(s) + u0(XT,x(0))γ(T, 0, x).

(10.7)

By direct computation,

E(E

(E(h)Y (T, x)|FWT

))= E (E(h)Y (T, x)) = E′Y (T, x),

where E′ is the expectation with respect to the measure dP′T = E(h)dPT and PT is therestriction of P to FW

T .

To proceed, let us first assume that the input data u0, f , and gk are all smooth functions withcompact support. Then, applying the Feynmann-Kac formula to the solution of equation(10.6) and using the Girsanov theorem (see, for example, Theorems 3.5.1 and 5.7.6 in [15]),we conclude that Uh(T, x) = E′Y (T, x) or

E(E(h)EY (t, x)|FW

T

)= E (E (h) U(T, x)) .

By Remark 7.8, the last equality implies U (T, ·) = E(Y (T, ·)|FW

T

)as elements of

L2

(Ω;L2,(r)(Rd)

).


To remove the additional smoothness assumption on the input data, let un0 , fn, and gn

k besequences of smooth compactly supported functions so that

limn→∞

(‖u0 − un

0‖2L2,(r)(Rd) +

∫ T

0‖f − fn‖2

L2,(r)(Rd)(t)dt

+∑

k≥1

∫ T

0q2k‖gk − gn

k‖2L2,(r)(Rd)(t)dt

)= 0.

(10.8)

Denote by Un and Y n the corresponding objects defined by (10.5) and (10.7) respectively.By Theorem 9.1, we have

(10.9) limn→∞E‖U − Un‖2

L2,(r)(Rd)(T ) = 0.

To complete the proof, it remains to show that

(10.10) limn→∞E

∥∥∥E(Y (T, ·)− Y n(T, ·)

∣∣∣FWT

)∥∥∥2

L2,(r)(Rd)= 0.

To this end, introduce a new probability measure P′′T by

dP′′T = exp

(2

∑

k≥1

∫ T

0νk(s,X

QT,x(s))qk

←−−dwk(s)

− 2∫ T

0

∑

k≥1

q2k|νk(s,X

QT,x(s))|2ds

)dPT .

By Girsanov’s theorem, equation (10.2) can be rewritten as

XT,x,i (s) = xi +∫ T

s

∑

k≥1

σik (τ,XT,x (τ))hk (τ) qkdτ

+∫ t

s(bi +

∑

k≥1

q2kσikνk) (τ,XT,x (τ)) dτ

+∫ t

s

∑

k≥1

qkσik (τ, XT,x (τ))←−−dw′′k (τ) +

∫ t

sσik (τ, XT,x (τ))

←−−dw′′k (τ) ,

(10.11)

where w′′k and w′′k are independent Winer processes with respect to the measure P′′T . Denoteby p(s, y|x) the corresponding distribution density of XT,x(s) and write `(x) = (1 + |x|2)r.It then follows by the Holder and Jensen inequalities that

E∥∥∥∥E

(∫ T

0γ2(T, s, ·)(f − fn)(s,XT,·(s))ds

∣∣∣FWT

)∥∥∥∥2

L2,(r)(Rd)

≤ K1

∫

Rd

(∫ T

0E

(γ2(T, s, x)(f − fn)2(s,XT,x(s))

)ds

)`(x)dx

≤ K2

∫

Rd

(∫ T

0E′′(f − fn)2(s,XT,x(s))ds

)`(x)dx

= K2

∫

Rd

∫ T

0

∫

Rd

(f(s, y)− fn(s, y))2p(s, y|x)dy ds `(x)dx,

(10.12)

where the number K1 depends only on T , and the number K2 depends only on T andsup(t,x) |c(t, x)| + ∑

k≥1 q2k sup(t,x) |νk(t, x)|2. Assumptions B0–B2 imply that there exist


positive numbers K3 and K4 so that

(10.13) p(s, y|x) ≤ K3

(T − s)d/2exp

(−K4

|x− y|2T − s

);

see, for example, [6]. As a result,∫

Rd

p(s, y|x)`(x)dx ≤ K5`(y),

and∫

Rd

∫ T

0

∫

Rd

(f(s, y)− fn(s, y))2p(s, y|x)dy ds `(x)dx

≤ K5

∫ T

0‖f − fn‖2

L2,(r)(Rd)(s)ds → 0, n →∞,

(10.14)

where the number K5 depends only on K3,K4, T , and r.

Calculations similar to (10.12)–(10.14) show that

E∥∥∥E

(γ2(T, 0, ·)(u0 − un

0 )(XT,·(0))∣∣∣W

)∥∥∥2

L2,(r)(Rd)

+ E

∥∥∥∥∥∥E

∫ T

0

∑

k≥1

(gk − gnk )(s,XT,·(s))γ(t, s, ·)qk

←−−dwk(s)

∣∣∣W

∥∥∥∥∥∥

2

L2,(r)(Rd)

→ 0(10.15)

as n → ∞. Then convergence (10.10) follows, which, together with (10.9), implies thatU (T, ·) = E

(UQ(T, ·)|FW

T

)as elements of L2

(Ω;L2,(r)(Rd)

). It remains to note that u =

Q−1U . Theorem 10.2 is proved. ¤

Given f ∈ L2,(r), we say that f ≥ 0 if and only if∫

Rd

f(x)ϕ(x)dx ≥ 0

for every non-negative ϕ ∈ C∞0 (Rd). Then Theorem 10.2 implies the following result.

Corollary 10.3. In addition to Assumptions B0–B5, let u0 ≥ 0, f ≥ 0, and gk = 0 forall k ≥ 1. Then u ≥ 0.

Proof. This follows from (10.3) and Proposition 7.11. ¤

Example 10.4. (Krylov-Veretennikov formula)

Consider the equation

(10.16) du = (aijDiDju + biDiu) dt +d∑

k=1

σikDiudwk, u (0, x) = u0 (x) .

Assume B0–B5 and suppose that aij(t, x) = 12σik(t, x)σjk(t, x). By Theorem 9.1, equation

(10.16) has a unique Wiener chaos solution so that

E‖u‖2L2(Rd)(t) ≤ C∗‖u0‖2

L2(Rd)


and

u (t, x) =∞∑

n=1

∑

|α|=n

uα(t, x)ξα = u0 (x) +∞∑

n=1

d∑

k1,...,kn=1

∫ t

0

∫ sn

0. . .

∫ s2

0

Φt,snσjknDj · · ·Φs2,s1σik1DiΦs1,0u0(x)dwk1(s1) · · · dwkn(sn),

(10.17)

where Φt,s is the semi-group generated by the operator A = aijDiDju + biDiu. On theother hand, in this case, Theorem 10.2 yields

u(t, x) = E

(u0(Xt,x(0))

∣∣FWt

),

where W = (w1, ..., wd) and

Xt,x,i (s) = xi +∫ t

sbi (τ, Xt,x (τ)) dτ +

d∑

k=1

σik (τ, Xt,x (τ))←−−dwk (τ)

s ∈ (0, t), t ∈ (0, T ], t− fixed.

(10.18)

Thus, we have arrived at the Krylov-Veretennikov formula [20, Theorem 4]

E(u0 (Xt,x (0)) |FW

t

)= u0 (x) +

∞∑

n=1

d∑

k1,...,kn=1

∫ t

0

∫ sn

0. . .

∫ s2

0

Φt,snσjknDj · · ·Φs2,s1σik1DiΦs1,0u0(x)dwk1(s1) · · · dwkn(sn).

(10.19)

11. Wiener Chaos and Nonlinear Filtering

In this section, we discuss some applications of the Wiener Chaos expansion to numeri-cal solution of the nonlinear filtering problem for diffusion processes; the presentation isessentially based on [25].

Let (Ω,F ,P) be a complete probability space with independent standard Wiener processesW = W (t) and V = V (t) of dimensions d1 and r respectively. Let X0 be a random variableindependent of W and V . In the diffusion filtering model, the unobserved d - dimensionalstate (or signal) process X = X(t) and the r-dimensional observation process Y = Y (t) aredefined by the stochastic ordinary differential equations

(11.1)dX(t) = b(X(t))dt + σ(X(t))dW (t) + ρ(X(t))dV (t),dY (t) = h(X(t))dt + dV (t), 0 < t ≤ T ;X(0) = X0, Y (0) = 0,

where b(x) ∈ Rd, σ(x) ∈ Rd×d1 , ρ(x) ∈ Rd×r, h(x) ∈ Rr.

Denote by Cn(Rd) the Banach space of bounded, n times continuously differentiable func-tions on Rd with finite norm

‖f‖Cn(Rd) = supx∈Rd

|f(x)|+ max1≤k≤n

supx∈Rd

|Dkf(x)|.

Assumption R1. The the components of the functions σ and ρ are in C2(Rd), the com-ponents of the functions b are in C1(R), the components of the function h are boundedmeasurable, and the random variable X0 has a density u0.


Assumption R2. The matrix σσ∗ is uniformly positive definite: there exists an ε > 0 sothat

d∑

i,j=1

d1∑

k=1

σik(x)σjk(x)yiyj ≥ ε|y|2, x, y ∈ Rd.

Under Assumption R1 system (11.1) has a unique strong solution [15, Theorems 5.2.5 and5.2.9]. Extra smoothness of the coefficients in assumption R1 insure the existence of aconvenient representation of the optimal filter.

If f = f(x) is a scalar measurable function on Rd so thatsup0≤t≤T E|f(X(t))|2 < ∞, then the filtering problem for (11.1) is to find the best meansquare estimate ft of f(X(t)), t ≤ T, given the observations Y (s), 0 < s ≤ t.

Denote by FYt the σ-algebra generated by Y (s), 0 ≤ s ≤ t. Then the properties of the

conditional expectation imply that the solution of the filtering problem is

ft = E(f(X(t))|FY

t

).

To derive an alternative representation of ft, some additional constructions will be necessary.

Define a new probability measure P on (Ω,F) as follows: for A ∈ F ,

P(A) =∫

AZ−1

T dP,

where

Zt = exp∫ t

0h∗(X(s))dY (s)− 1

2

∫ t

0|h(X(s))|2ds

(here and below, if ζ ∈ Rk, then ζ is a column vector, ζ∗ = (ζ1, . . . , ζk), and |ζ|2 = ζ∗ζ).If the function h is bounded, then the measures P and P are equivalent. The expectationwith respect to the measure P will be denoted by E.

The following properties of the measure P are well known [14, 42]:

P1. Under the measure P, the distributions of the Wiener process W and the randomvariable X0 are unchanged, the observation process Y is a standard Wiener process,and, for 0 < t ≤ T , the state process X satisfies

dX(t) = b(X(t))dt + σ(X(t))dW (t) + ρ(X(t)) (dY (t)− h(X(t))dt) ,X(0) = X0;

P2. Under the measure P, the Wiener processes W and Y and the random variable X0

are independent of one another;P3. The optimal filter ft satisfies

(11.2) ft =E

[f(X(t))Zt|FY

t

]

E[Zt|FYt ]

.

Because of property P2 of the measure P the filtering problem will be studied on theprobability space (Ω,F , P). In particular, we will consider the stochastic basis F =Ω,F , FY

t 0≤t≤T , P and the Wiener Chaos space L2(Y) of FYT -measurable random vari-

ables η with E|η|2 < ∞.


If the function h is bounded, then, by the Cauchy-Schwarz inequality,

(11.3) E|η| ≤ C(h, T )√E|η|2, η ∈ L2(Y).

Next, consider the partial differential operators

Lg(x) =12

d∑

i,j=1

((σ(x)σ∗(x))ij + (ρ(x)ρ∗(x))ij)∂2g(x)∂xi∂xj

+d∑

i=1

bi(x)∂g(x)∂xi

;

Mlg(x) = hl(x)g(x) +d∑

i=1

ρil(x)∂g(x)∂xi

, l = 1, . . . , r;

and their adjoints

L∗g(x) =12

d∑

i,j=1

∂2

∂xi∂xj((σ(x)σ∗(x))ijg(x) + (ρ(x)ρ∗(x))ijg(x))

−d∑

i=1

∂

∂xi(bi(x)g(x)) ;

M∗l g(x) = hl(x)g(x)−

d∑

i=1

∂

∂xi(ρil(x)g(x)) , l = 1, . . . , r.

Note that, under the assumptions R1 and R2, the operators L,L∗ are bounded from H12 (Rd)

to H−12 (Rd), operators M,M∗ are bounded from H1

2 (Rd) to L2(Rd), and

(11.4) 2〈L∗v, v〉+r∑

l=1

‖M∗l v‖2

L2(Rd) + ε‖v‖2H2

1 (Rd) ≤ C‖v‖2L2(Rd), v ∈ H1

2 (Rd),

where 〈·, ·〉 is the duality between H12Rd and H−1

2 (Rd). The following result is well known[42, Theorem 6.2.1].

Proposition 11.1. In addition to Assumptions R1 and R1 suppose that the initial densityu0 belongs to L2(Rd). Then there exists a random field u = u(t, x), t ∈ [0, T ], x ∈ Rd, withthe following properties:

1. u ∈ L2(Y;L2((0, T );H12 (Rd))) ∩ L2(Y;C([0, T ], L2(Rd))).

2. The function u(t, x) is a traditional solution of the stochastic partial differential equation

(11.5)du(t, x) = L∗u(t, x)dt +

r∑

l=1

M∗l u(t, x)dYl(t), 0 < t ≤ T, x ∈ Rd;

u(0, x) = u0(x).

3. The equality

(11.6) E[f(X(t))Zt|FY

t

]=

∫

Rd

f(x)u(t, x)dx

holds for all bounded measurable functions f .

The random field u = u(t, x) is called the unnormalized filtering density (UFD) and therandom variable φt[f ] = E

[f(X(t))Zt|FY

t

], the unnormalized optimal filter.


A number of authors studied the nonlinear filtering problem using the multiple Ito integralversion of the Wiener chaos [2, 21, 39, 46, etc.]. In what follows, we construct approximationsof u and φt[f ] using the Cameron-Martin version.

By Theorem 8.6,

(11.7) u(t, x) =∑

α∈Juα(t, x)ξα,

where

(11.8) ξα =1√α!

∏

i,k

Hαki(ξik), ξik =

∫ T

0mi(t)dYk(t), k = 1, . . . , r;

as before, Hn(·) is the Hermite polynomial (3.3) and mi ∈ m, an orthonormal basis inL2((0, T )). The functions uα satisfy the corresponding propagator

∂

∂tuα(t, x) = L∗uα(t, x)

+∑

k,i

√αk

iM∗kuα−(i,k)(t, x)mi(t), 0 < t ≤ T, x ∈ Rd;

u(0, x) = u0(x)I(|α| = 0).

(11.9)

Writing

fα(t) =∫

Rd

f(x)uα(t, x)dx,

we also get a Wiener chaos expansion for the unnormalized optimal filter:

(11.10) φt[f ] =∑

α∈Jfα(t)ξα, t ∈ [0, T ].

For a positive integer N , define

(11.11) uN (t, x) =∑

|α|≤N

uα(t, x)ξα.

Theorem 11.2. Under Assumptions R1 and R2, there exists a positive number ν, depend-ing only on the functions h and ρ, so that

(11.12) E‖u− uN‖2L2(Rd)(t) ≤

‖u0‖2L2(Rd)

ν(1 + ν)N, t ∈ [0, T ].

If, in addition, ρ = 0, then there exists a real number C, depending only on the functions band σ, so that

(11.13) E‖u− uN‖2L2(Rd)(t) ≤

(4h∞t)N+1

(N + 1)!eCt‖u0‖2

L2(Rd), t ∈ [0, T ],

where h∞ = maxk=1,...,r supx |hk(x)|.

For positive integers N, n, define a set of multi-indices

J nN = α = (αk

i , k = 1, . . . , r, i = 1, . . . , n) : |α| ≤ N.and let

(11.14) unN (t, x) =

∑

α∈J nN

uα(t, x)ξα.


Unlike Theorem 11.2, to compute the approximation error in this case we need to choosea special basis m — to do the error analysis for the Fourier approximation in time. Wealso need extra regularity of the coefficients in the state and observation equations — tohave the semi-group generated by the operator L∗ continuous not only in L2(Rd) but alsoin H2

2 (Rd). The resulting error bound is presented below; the proof can be found in [25].

Theorem 11.3. Assume that

(1) The basis m is the Fourier cosine basis

(11.15) m1(s)=1√T

; mk(t)=

√2T

cos(

π(k − 1)tT

), k > 1; 0 ≤ t ≤ T,

(2) The components of the functions σ are in C4(Rd), the components of the functions bare in C3(R), the components of the function h are in C2(Rd); ρ = 0; u0 ∈ H2

2 (Rd).

Then there exist a positive number B1 and a real number B2, both depending only on thefunctions b and σ so that

(11.16) E‖u− unN‖2

L2(Rd)(T )≤B1eB2T

((4h∞T )N+1

(N + 1)!eCt‖u0‖2

L2(Rd) +T 3

n‖u0‖2

H22 (Rd)

),

where h∞ = maxk=1,...,r supx |hk(x)|.

12. Passive Scalar in a Gaussian Field

This section presents the results from [29] and [28] about the stochastic transport equation.

The following viscous transport equation is used to describe time evolution of a scalarquantity θ in a given velocity field v:

(12.1) θ(t, x) = ν∆θ(t, x)− v(t, x) · ∇θ(t, x) + f(t, x); x ∈ Rd, d > 1.

The scalar θ is called passive because it does not affect the velocity field v.

We assume that v = v(t, x) ∈ Rd is an isotropic Gaussian vector field with zero mean andcovariance

E(vi(t, x)vj(s, y)) = δ(t− s)Cij(x− y),where C = (Cij(x), i, j = 1, . . . , d) is a matrix-valued function so that C(0) is a scalarmatrix; with no loss of generality we will assume that C(0) = I, the identity matrix.

It is known from [22, Section 10.1] that, for an isotropic Gaussian vector field, the Fouriertransform C = C(z) of the function C = C(x) is

(12.2) C(y) =A0

(1 + |y|2)(d+α)/2

(ayy∗

|y|2 +b

d− 1

(I − yyT

|y|2))

,

where y∗ is the row vector (y1, . . . , yd), y is the corresponding column vector, |y|2 = y∗y;γ > 0, a ≥ 0, b ≥ 0, A0 > 0 are real numbers. Similar to [22], we assume that 0 < γ < 2.This range of values of γ corresponds to a turbulent velocity field v, also known as thegeneralized Kraichnan model [8]; the original Kraichnan model [18] corresponds to a = 0.For small x, the asymptotics of Cij(x) is (δij − cij |x|γ) [22, Section 10.2].

By direct computation (cf. [1]), the vector field v = (v1, . . . , vd) can be written as

(12.3) vi(t, x) = σik(x)wk(t),


where σk, k ≥ 1 is an orthonormal basis in the space HC , the reproducing kernel Hilbertspace corresponding to the kernel function C. It is known from [22] that HC is all or partof the Sobolev space H(d+γ)/2(Rd;Rd).

If a > 0 and b > 0, then the matrix C is invertible and

HC =

f ∈ Rd :∫

Rd

f∗(y)C−1(y)f(y)dy < ∞

= H(d+γ)/2(Rd;Rd),

because ‖C(y)‖ ∼ (1 + |y|2)−(d+γ)/2.

If a > 0 and b = 0, then

HC =

f ∈ Rd :∫

Rd

|f(y)|2(1 + |y|2)(d+γ)/2dy < ∞; yy∗f(y) = |y|2f(y)

,

the subset of gradient fields in H(d+γ)/2(Rd;Rd), that is, vector fields f for which f(y) =yF (y) for some scalar F ∈ H(d+γ+2)/2(Rd).

If a = 0 and b > 0, then

HC =

f ∈ Rd :∫

Rd

|f(y)|2(1 + |y|2)(d+γ)/2dy < ∞; y∗f(y) = 0

,

the subset of divergence-free fields in H(d+γ)/2(Rd;Rd).

By the embedding theorems, each σik is a bounded continuous function on Rd; in fact, every

σik is Holder continuous of order γ/2. In addition, being an element of the corresponding

space HC , each σk is a gradient field if b = 0 and is divergence free if a = 0.

Equation (12.1) becomes

(12.4) dθ(t, x) = (ν∆θ(t, x) + f(t, x))dt−∑

k

σk(x) · ∇θ(t, x)dwk(t).

We summarize the above constructions in the following assumptions:

S1 There is a fixed stochastic basis F = (Ω,F , Ftt≥0,P) with the usual assumptionsand (wk(t), k ≥ 1, t ≥ 0) is a collection of independent standard Wiener processeson F.

S2 For each k, the vector field σk is an element of the Sobolev spaceH

(d+γ)/22 (Rd;Rd), 0 < γ < 2, d ≥ 2.

S3 For all x, y in Rd,∑

k σik(x)σj

k(y) = Cij(x − y) so that the matrix-valued functionC = C(x) satisfies (12.2) and C(0) = I.

S4 The input data θ0, f are deterministic and satisfy

θ0 ∈ L2(Rd), f ∈ L2((0, T );H−12 (Rd));

ν > 0 is a real number.

Theorem 12.1. Let Q be a sequence with qk = q <√

2ν, k ≥ 1.

Under assumptions S1–S4, there exits a unique w(H12 (Rd),H−1

2 (Rd)) Wiener Chaos solu-tion of (12.4). This solution is an FW

t -adapted process and satisfies

‖θ‖2L2,Q(W;L2((0,T );H1

2 (Rd))) + ‖θ‖2L2,Q(W;C((0,T );L2(Rd)))

≤ C(ν, q, T )(‖θ0‖2

L2(Rd) + ‖f‖2L2((0,T );H−1

2 (Rd))

).


Theorem 12.1 provides new information about the solution of equation (12.1) for all valuesof ν > 0. Indeed, if

√2ν > 1, then q > 1 is an admissible choice of the weights, and, by

Proposition 7.4(1), the solution θ has Malliavin derivatives of every order. If√

2ν ≤ 1, thenequation (12.4) does not have a square-integrable solution.

Note that if the weight is chosen so that q =√

2ν, then equation (12.1) can still be analyzedusing Theorem 9.1 in the normal triple (H1

2 (Rd), L2(Rd), H−12 (Rd)).

If ν = 0, equation (12.4) must be interpreted in the sense of Stratonovich:

(12.5) du(t, x) = f(t, x)dt− σk(x) · ∇θ(t, x) dwk(t).

To simplify the presentation, we assume that f = 0. If (12.2) holds with a = 0, then eachσk is divergence free and (12.5) has an equivalent Ito form

(12.6) dθ(t, x) =12∆θ(t, x)dt− σi

k(x)Diθ(t, x)dwk(t).

Equation (12.6) is a model of non-viscous turbulent transport [5]. The propagator for (12.6)is

(12.7)∂

∂tθα(t, x) =

12∆θα(t, x)−

∑

i,k

√αk

i σjkDjθα−(i,k)(t, x)mi(t), 0 < t ≤ T,

with initial condition θα(0, x) = θ0(x)I(|α| = 0).

The following result about solvability of (12.6) is proved in [29] and, in a slightly weakerform, in [28].

Theorem 12.2. In addition to S1–S4, assume that each σk is divergence free. Then thereexits a unique w(H1

2 (Rd),H−12 (Rd)) Wiener Chaos solution θ = θ(t, x) of (12.6). This

solution has the following properties:

(A) For every ϕ ∈ C∞0 (Rd) and all t ∈ [0, T ], the equality

(12.8) (θ, ϕ)(t) = (θ0, ϕ) +12

∫ t

0(θ, ∆ϕ)(s)ds +

∫ t

0(θ, σi

kDiϕ)dwk(s)

holds in L2(FWt ), where (·, ·) is the inner product in L2(Rd).

(B) If X = Xt,x is a weak solution of

(12.9) Xt,x = x +∫ t

0σk (Xs,x) dwk (s) ,

then, for each t ∈ [0, T ],

(12.10) θ (t, x) = E(θ0 (Xt,x) |FW

t

).

(C) For 1 ≤ p < ∞ and r ∈ R, define Lp,(r)(Rd) as the Banach space of measurable functionswith norm

‖f‖pLp,(r)(Rd)

=∫

Rd

|f(x)|p(1 + |x|2)pr/2dx

is finite. Then there exits a number K depending only on p, r so that, for each t > 0,

(12.11) E‖θ‖pLp,(r)(Rd)

(t) ≤ eKt‖θ0‖pLp,(r)(Rd)

.

In particular, if r = 0, then K = 0.


It follows that, for all s, t and almost all x, y,

Eθ (t, x) = θα (t, x) I|α|=0

and

Eθ (t, x) θ (s, y) =∑

α∈Jθα (t, x) θα (s, y) .

If the initial condition θ0 belongs to L2(Rd) ∩ Lp(Rd) for p ≥ 3, then, by (12.11), higherorder moments of θ exist. To obtain the expressions of the higher-order moments in termsof the coefficients θα, we need some auxiliary constructions.

For α, β ∈ J , define α+β as the multi-index with components αki +βk

i . Similarly, we definethe multi-indices |α− β| and α ∧ β = min(α, β). We write β ≤ α if and only if βk

i ≤ αki for

all i, k ≥ 1. If β ≤ α, we define(

α

β

):=

∏

i,k

αki !

βki !(αk

i − βki )!

.

Definition 12.3. We say that a triple of multi-indices (α, β, γ) is complete and write(α, β, γ) ∈ 4 if all the entries of the multi-index α + β + γ are even numbers and|α− β| ≤ γ ≤ α + β. For fixed α, β ∈ J , we write

4 (α) := γ, µ ∈ J : (α, γ, µ) ∈ 4and

4(α, β) := γ ∈ J : (α, β, γ) ∈ 4.

For (α, β, γ) ∈ 4, we define

(12.12) Ψ (α, β, γ) :=√

α!β!γ!((

α− β + γ

2

)!(

β − α + γ

2

)!(

α + β − γ

2

)!)−1

.

Note that the triple (α, β, γ) is complete if and only if any permutation of the triple (α, β, γ)is complete. Similarly, the value of Ψ (α, β, γ) is invariant under permutation of the argu-ments.

We also define

(12.13) C (γ, β, µ) :=[(

γ + β − 2µ

γ − µ

)(γ

µ

)(β

µ

)]1/2

, µ ≤ γ ∧ β.

It is readily checked that if f is a function on J , then for γ, β ∈ J ,

(12.14)∑

µ≤γ∧β

C (γ, β, p) f (γ + β − 2µ) =∑

µ∈(γ,β)

f (µ)Φ (γ, β, µ)

The next theorem presents the formulas for the third and fourth moments of the solutionof equation (12.6) in terms of the coefficients θα.

Theorem 12.4. In addition to S1–S4, assume that each σk is divergence free and theinitial condition θ0 belongs to L2(Rd) ∩ L4(Rd). Then

(12.15) Eθ(t, x)θ(t′, x′

)θ(s, y) =

∑

(α,β,γ)∈4Ψ(α, β, γ) θα(t, x)θβ(t′, x′)θγ (s, y)


and

Eθ(t, x)θ(t′, x′)θ (s, y) θ(s′, y′

)(12.16)

=∑

ρ∈4(α,β)∩4(γ,κ)

Ψ(α, β, ρ)Ψ (ρ, γ, κ) θα (t, x) θβ(t′, x′)θγ (s, y) θκ

(s′, y′

).

Proof. It is known [30] that

(12.17) ξγξβ =∑

µ≤γ∧β

C (γ, β, µ) ξγ+β−2µ.

Let us consider the triple product ξαξβξγ . By (12.17),

(12.18) Eξαξβξγ = E∑

µ∈4(α,β)

ξγξµΨ(α, β, µ) =

Ψ(α, β, γ) , (α, β, γ) ∈ 4;0, otherwise.

Equality (12.15) now follows.

To compute the fourth moment, note that

ξαξβξγ =∑

µ≤α∧β

C (α, β, µ) ξα+β−2µξγ

=∑

µ≤α∧β

C (α, β, µ)∑

ρ≤(α+β−2µ)∧γ

C (α + β − 2µ, γ, ρ) ξα+β+γ−2µ−2ρ.(12.19)

Repeated applications of (12.14) yield

ξαξβξγ =∑

µ≤α∧β

C (α, β, µ)∑

ρ∈4(α+β−2µ,γ)

ξρΨ(α + β − 2µ, γ, ρ)

=∑

µ∈4(α,β)

∑

ρ∈4(µ,γ)

Ψ (α, β, µ)Ψ (µ, γ, ρ) ξρ

Thus,

Eξαξβξγξκ =∑

µ∈4(α,β)

∑

ρ∈4(µ,γ)

Ψ (α, β, µ)Ψ (µ, γ, ρ) Iµ=κ

=∑

ρ∈4(α,β)∩4(γ,κ)

Ψ(α, β, ρ) Ψ (ρ, γ, κ) .

Equality (12.16) now follows. ¤

In the same way, one can get formulas for fifth- and higher-order moments.

Remark 12.5. Expressions (12.15) and (12.16) do not depend on the structure of equation(12.6) and can be used to compute the third and fourth moments of any random field witha known Cameron-Martin expansion. The interested reader should keep in mind that theformulas for the moments of orders higher then two should be interpreted with care. In fact,they represent the pseudo-moments (for detail see [35]).

We now return to the analysis of the passive scalar equation (12.4). By reducing thesmoothness assumptions on σk, it is possible to consider velocity fields v that are moreturbulent than in the Kraichnan model, for example,

(12.20) vi(t, x) =∑

k≥0

σik(x)wk(t),


where σk, k ≥ 1 is an orthonormal basis in L2(Rd;Rd). With v as in (12.20), the passivescalar equation (12.4) becomes

(12.21) θ(t, x) = ν∆θ(t, x) + f(t, x)−∇θ(t, x) · W (t, x),

where W = W (t, x) is a d-dimensional space-time white noise and the Ito stochastic differ-ential is used. Previously, such equations have been studied using white noise approach inthe space of Hida distributions [4, 40]. A summary of the related results can be found in[12, Section 4.3].

The Q-weighted Wiener chaos spaces allow us to state a result that is fully analogous toTheorem 12.1. The proof is derived from Theorem 9.1; see [29] for details.

Theorem 12.6. Suppose that ν > 0 is a real number, each |σik(x)| is a bounded mea-

surable function, and the input data are deterministic and satisfy u0 ∈ L2(Rd), f ∈L2

((0, T );H−1

2 (Rd)).

Fix ε > 0 and let Q = qk, k ≥ 1 be a sequence so that, for all x, y ∈ Rd,

2ν|y|2 −∑

k≥1

q2kσ

ik(x)σj

k(x)yiyj ≥ ε|y|2.

Then, for every T > 0, there exits a unique w(H12 (Rd),H−1

2 (Rd)) Wiener Chaos solution θof equation

(12.22) dθ(t, x) = (ν∆θ(t, x) + f(t, x))dt− σk(x) · ∇θ(t, x)dwk(t),

The solution is an Ft-adapted process and satisfies

‖θ‖2L2,Q(W;L2((0,T );H1

2 (Rd))) + ‖θ‖2L2,Q(W;C((0,T );L2(Rd)))

≤ C(ν, q, T )(‖θ0‖2

L2(Rd) + ‖f‖2L2((0,T );H−1

2 (Rd))

).

If maxi supx |σik(x)| ≤ Ck, k ≥ 1, then a possible choice of Q is

qk = (δν)1/2/(d2kCk), 0 < δ < 2.

If σik(x)σj

k(x) ≤ Cσ < +∞, i, j = 1, . . . , d, x ∈ Rd, then a possible choice of Q is

qk = ε (2ν/(Cσd))1/2 , 0 < ε < 1.

13. Stochastic Navier-Stokes Equation

In this section, we review the main facts about the stochastic Navier-Stokes equation andindicate how the Wiener Chaos approach can be used in the study of non-linear equations.Most of the results of this section come from the two papers [35] and [31].

A priori, it is not clear in what sense the motion described by Kraichnan’s velocity (seeSection 12) might fit into the paradigm of Newtonian mechanics. Accordingly, relatingthe Kraichnan velocity field v to classic fluid mechanics naturally leads to the questionwhether we can compensate v (t, x) by a field u (t, x) that is more regular with respect tothe time variable, so that there is a balance of momentum for the resulting field U (t, x) =u (t, x) + v (t, x) or, equivalently, that the motion of a fluid particle in the velocity fieldU (t, x) satisfies the Second Law of Newton.


A positive answer to this question is given in [35], where it is shown that the equation forthe smooth component u = (u1, . . . , ud) of the velocity is given by

(13.1)

dui = [ν∆ui − ujDjui −DiP + fi]dt

+(gik −DiPk −Djσ

jku

i)

dwk, i = 1, . . . , d, 0 < t ≤ T ;

divu = 0, u(0, x) = u0(x).

where wk, k ≥ 1 are independent standard Wiener processes on a stochastic basis F,the functions σj

k are given by (12.3), the known functions f = (f1, . . . , fd), gk = (gik),

i = i, . . . , d, k ≥ 1 are, respectively, the drift and the diffusion components of the free force,and the unknown functions P , Pk are the drift and diffusion components of the pressure.

Remark 13.1. It is useful to study equation (13.1) for more general coefficients σjk. So, in

the future, σjk are not necessarily the same as in Section 12.

We make the following assumptions:

NS1 The functions σik = σi

k(t, x) are deterministic and measurable,

∑

k≥1

(d∑

i=1

|σik(t, x)|2 + |Diσ

ik(t, x)|2

)≤ K,

and there exists ε > 0 so that, for all y ∈ Rd,

ν|y|2 − 12σi

k(t, x)σjk(t, x)yiyj ≥ ε|y|2,

t ∈ [0, T ], x ∈ Rd.NS2 The functions f i, gi

k are non-random and

d∑

i=1

‖f i‖2

L2((0,T );H−12 (Rd))

+∑

k≥1

‖gik‖2

L2((0,T );L2(Rd))

< ∞.

Remark 13.2. In NS1, the derivatives Diσik are understood as Schwartz distributions, but

it is assumed that divσ :=∑d

i=1 ∂iσi is a bounded l2−valued function. Obviously, the latter

assumption holds in the important case when∑d

i=1 ∂iσi = 0.

Our next step is to use the divergence-free property of u to eliminate the pressure P andP from equation (13.1). For that, we need the decomposition of L2(Rd;Rd) into potentialand solenoidal components.

Write S(L2(Rd;Rd)) = V ∈ L2(Rd;Rd) : div V = 0. It is known (see e.g. [16]) that

L2(Rd;Rd) = G(L2(Rd;Rd))⊕S(L2(Rd;Rd)),

where G(L2(Rd;Rd)) is a Hilbert subspace orthogonal to S(L2(Rd;Rd)).

The functions G(V) and S(V) can be defined for V from any Sobolev space Hγ2 (Rd;Rd) and

are usually referred to as the potential and the divergence free (or solenoidal), projections,respectively, of the vector field V.

Now let u be a solution of equation (13.1). Since div u = 0, we have

Di(ν∆ui − ujDjui −DiP + f i) = 0; Di(σ

jkDju

jui + gik −DiPk) = 0, k ≥ 1.


As a result,

DiP = G(ν∆ui − ujDjui + f i); DiPk = G(σj

kDjui + gi

k), i = 1, . . . , d, k ≥ 1.

So, instead of equation (13.1), we can and will consider its equivalent form for the unknownvector u = (u1, . . . , ud):

(13.2) du = S(ν∆u− ujDju + f)dt + S(σjkDju + gk)dwk, 0 < t ≤ T,

with initial condition u|t=0 = u0.

Definition 13.3. An Ft-adapted random process u from the space L2(Ω×[0, T ];H12 (Rd;Rd))

is called a solution of equation (13.2) if

(1) With probability one, the process u is weakly continuous in L2(Rd;Rd).(2) For every ϕ ∈ C∞

0 (Rd,Rd), with div ϕ = 0 there exists a measurable set Ω′ ⊂ Ω sothat, for all t ∈ [0, T ], the equality

(ui, ϕi)(t) = (ui0, ϕ

i) +∫ t

0

((νDju

i, Djϕi)(s) + 〈f i, ϕi〉(s))ds

∫ t

0

(σj

kDjui + gi, ϕi)dwk(s)

(13.3)

holds on Ω′. In (13.3), (·, ·) is the inner product in L2(Rd) and 〈·, ·, 〉 is the dualitybetween H1

2 (Rd) and H−12 (Rd).

The following existence and uniqueness result is proved in [31].

Theorem 13.4. In addition to NS1 and NS2, assume that the initial condition u0 is non-random and belongs to L2(Rd;Rd). Then there exist a stochastic basis F = (Ω,F , Ftt≥0,P)with the usual assumptions, a collection wk, k ≥ 1 of independent standard Wiener pro-cesses on F, and a process u so that u is a solution of (13.2) and

E

(sups≤T

‖u(s)‖2L2(Rd;Rd) +

∫ T

0‖∇u(s)‖2

L2(Rd;Rd) ds

)< ∞.

If, in addition, d = 2, then the solution of (13.2) exists on any prescribed stochastic basis, isstrongly continuous in t, is FW

t -adapted, and is unique, both path-wise and in distribution.

When d ≥ 3, existence of a strong solution as well as uniqueness (strong or weak) forequation (13.2) are important open problems.

By the Cameron-Martin theorem,

u(t, x) =∑

α∈Juα(t, x)ξα.

If the solution of (13.2) is FWt -adapted, then, using the Ito formula together with relation

(5.5) for the time evolution of E(ξα|FWt ) and relation (12.17) for the product of two elements

of the Cameron-Martin basis, we can derive the propagator system for coefficients uα [31,Theorem 3.2]:

Theorem 13.5. In addition to NS1 and NS2, assume that u0 ∈ L2(Rd;Rd) and equation(13.2) has an FW

t -adapted solution u so that

(13.4) supt≤T

E‖u‖2L2(Rd;Rd)(t) < ∞.


Then

(13.5) u (t, x) =∑

α∈Juα (t, x) ξα,

and the Hermite-Fourier coefficients uα(t, x) are L2(Rd;Rd)-valued weakly continuous func-tions so that

(13.6) supt≤T

∑

α∈J‖uα‖2

L2(Rd;Rd)(t) +∫ T

0

∑

α∈J‖∇uα‖2

L2(Rd;Rd×d)(t) dt < ∞.

The functions uα (t, x) , α ∈ J , satisfy the (nonlinear) propagator

∂

∂tuα = S

(∆uα −

∑

γ,β∈∆(α)

Ψ(α, β, γ) (uγ ,∇uβ) + I|α|=0f

+∑

j,k

√αk

j

((σk,∇

)uα−(j,k) + I|α|=1gk

)mj (t)

), 0 < t ≤ T ;

uα|t=0 = u0I|α|=0;

(13.7)

recall that the numbers Ψ(α, β, γ) are defined in (12.12).

One of the questions in the theory of the Navier-Stokes equation is computation of the meanvalue u = Eu of the solution. The traditional approach relies on the Reynolds equation forthe mean

(13.8) ∂tu− ν∆u + ( u,∇) u = 0,

which is not really an equation with respect to u. Decoupling (13.8) has been an area ofactive research: Reynolds approximations, coupled equations for the moments, Gaussianclosures, and so on (see e.g. [36], [45] and the references therein)

Another way to compute u (t, x) is to find the distribution of v (t, x) using the infinite-dimensional Kolmogorov equation associated with (13.2). The complexity of this Kol-mogorov equation is prohibitive for any realistic application, at least for now.

The propagator provides a third way: expressing the mean and other statistical momentsof u in terms of uα. Indeed, by Cameron-Martin Theorem,

Eu(t, x) = u0(t, x),

Eui(t, x)uj (s, y) =∑

α∈Jui

α(t, x)ujα(s, y)

If exist, the third- and fourth-order moments can be computed using (12.15) and (12.16).

The next theorem, proved in [31], shows that the existence of a solution of the propagator(13.7) is not only necessary but, to some extent, sufficient for the global existence of aprobabilistically strong solution of the stochastic Navier-Stokes equation (13.2).

Theorem 13.6. Let NS1 and NS2 hold and u0 ∈ L2(Rd;Rd). Assume that the propagator(13.7) has a solution uα (t, x) , α ∈ J on the interval (0, T ] so that, for every α, theprocess uα is weakly continuous in L2(Rd;Rd) and the inequality

(13.9) supt≤T

∑

α∈J‖uα‖2

L2(Rd;Rd)(t) +∫ T

0

∑

α∈J‖∇uα‖2

L2(Rd;Rd×d)(t) dt < ∞


holds. If the process

(13.10) U (t, x) :=∑

α∈Juα (t, x) ξα

is FWt -adapted, then it is a solution of (13.2).

The process U satisfies

E

(sups≤T

‖U(s)‖2L2(Rd;Rd) +

∫ T

0‖∇U(s)‖2

L2(Rd;Rd×d) ds

)< ∞

and, for every v ∈ L2(Rd;Rd), E(U,v

)is a continuous function of t.

Since U is constructed on a prescribed stochastic basis and over a prescribed time interval[0, T ], this solution of (13.2) is strong in the probabilistic sense and is global in time.Being true in any space dimension d, Theorem 13.6 suggests another possible way to studyequation (13.2) when d ≥ 3. Unlike the propagator for the linear equation, the system(13.7) is not lower-triangular and not solvable by induction, so that analysis of (13.7) is anopen problem.

14. First-Order Ito Equations

The objective of this section is to study equation

(14.1) du(t, x) = ux(t, x)dw(t), t > 0, x ∈ R,

and its analog for x ∈ Rd.

Equation (14.1) was first encountered in Example 6.8; see also [9]. With a non-random initialcondition u(0, x) = ϕ(x), direct computations show that, if exists, the Fourier transformu = u(t, y) of the solution must satisfy

(14.2) du(t, y) =√−1yu(t, y)dw(t), or u(t, y) = ϕ(y)e

√−1yw(t)+ 12y2t.

The last equality shows that the properties of the solution essentially depend on the initialcondition, and, in general, the solution is not in L2(W).

The S-transformed equation, vt = h(t)vx, has a unique solution

v(t, x) = ϕ

(x +

∫ t

0h(s)ds

), h(t) =

N∑

i=1

himi(t).

The results of Section 3 imply that a white noise solution of the equation can exist only ifϕ is a real analytic function. On the other hand, if ϕ is infinitely differentiable, then, byTheorem 8.4, the Wiener Chaos solution exists and can be recovered from v.

Theorem 14.1. Assume that the initial condition ϕ belongs to the Schwarz space S = S(R)of tempered distributions. Then there exists a generalized random process u = u(t, x), t ≥ 0,x ∈ R, so that, for every γ ∈ R and T > 0, the process u is the unique w(Hγ

2 (R),Hγ−12 (R))

Wiener Chaos solution of equation (14.1).

Proof. The propagator for (14.1) is

(14.3) uα(t, x) = ϕ(x)I(|α| = 0) +∫ t

0

∑

i

√αi(uα−(i)(s, x))xmi(s)ds.


Even though Theorem 6.4 is not applicable, the system can be solved by induction if ϕis sufficiently smooth. Denote by Cϕ(k), k ≥ 0, the square of the L2(R) norm of the kth

derivative of ϕ:

(14.4) Cϕ(k) =∫ +∞

−∞|ϕ(k)(x)|2dx.

By Corollary 6.6, for every k ≥ 0 and n ≥ 0,

(14.5)∑

|α|=k

‖(u(n)α )x‖2

L2(R)(t) =tkCϕ(n + k)

k!.

The statement of the theorem now follows. ¤

Remark 14.2. Once interpreted in a suitable sense, the Wiener Chaos solution of (14.1)is FW

t -adapted and does not depend on the choice of the Cameron-Martin basis in L2(W).Indeed, choose the wight sequence so that

r2α =

11 + Cϕ(|α|) .

By (14.5), we have u ∈ RL2(W; L2(R)).

Next, define

ψN (x) =1π

sin(Nx)x

.

Direct computations show that the Fourier transform of ψN is supported in [−N, N ] and∫R ψN (x)dx = 1. Consider equation (14.1) with initial condition

ϕN (x) =∫

Rϕ(x− y)ψN (y)dy.

By (14.2), this equation has a unique solution uN so that uN (t, ·) ∈ L2(W; Hγ2 (R)), t ≥ 0,

γ ∈ R. Relation (14.5) and the definition of uN imply

limN→∞

∑

|α|=k

‖uα − uN,α‖2L2(R)(t) = 0, t ≥ 0, k ≥ 0,

so that, by the Lebesgue dominated convergence theorem,

limN→∞

‖u− uN‖2RL2(W;L2(R))(t) = 0, t ≥ 0.

In other words, the solution of the propagator (14.3) corresponding to any basis m inL2((0, T )) is a limit in RL2(W; L2(R)) of the sequence uN , N ≥ 1 of FW

t -adapted pro-cesses.

The properties of the Wiener Chaos solution of (14.1) depend on the growth rate of thenumbers Cϕ(n). In particular,

• If Cϕ(n) ≤ Cn(n!)γ , C > 0, 0 ≤ γ < 1, thenu ∈ L2 (W; L2((0, T );Hn

2 (R))) for all T > 0 and every n ≥ 0.• If Cϕ(n) ≤ Cnn!, C > 0, then

– for every n ≥ 0, there is a T > 0 so that u ∈ L2 (W;L2((0, T );Hn2 (R))). In

other words, the square-integrable solution exists only for sufficiently small T .– for every n ≥ 0 and every T > 0, there exists a number δ ∈ (0, 1) so that

u ∈ L2,Q (W; L2((0, T );Hn2 (R))) with Q = (δ, δ, δ, . . .).


• If the numbers Cϕ(n) grow as Cn(n!)1+ρ, ρ ≥ 0, then, for every T > 0, there existsa number γ > 0 so thatu ∈ (S)−ρ,−γ (L2(W);L2((0, T );Hn

2 (R))). If ρ > 0, then this solution does notbelong to any L2,Q (W; L2((0, T );Hn

2 (R))). If ρ > 1, then this solution does nothave an S-transform.

• If the numbers Cϕ(n) grow faster than Cn(n!)b for any b, C > 0, then the WienerChaos solution of (14.1) does not belong to any(S)−ρ,−γ (L2((0, T );Hn

2 (R))), ρ, γ > 0, or L2,Q (W; L2((0, T );Hn2 (R))).

To construct a function ϕ with the required rate of growth of Cϕ(n), consider

ϕ(x) =∫ ∞

0cos(xy)e−g(y)dy,

where g is a suitable positive, unbounded, even function. Note that, up to a multi-plicative constant, the Fourier transform of ϕ is e−g(y), and so Cϕ(n) grows with n as∫ +∞0 |y|2ne−2g(y)dy.

A more general first-order equation can be considered:

(14.6) du(t, x) = σik(t, x)Diu(t, x)dwk(t), t > 0, x ∈ Rd.

Theorem 14.3. Assume that in equation (14.6) the initial condition u(0, x) belongs toS(Rd) and each σik is infinitely differentiable with respect to x so that sup(t,x) |Dnσik(t, x)| ≤Cik(n), n ≥ 0. Then there exists a generalized random process u = u(t, x), t ≥ 0, x ∈ Rd, sothat, for every γ ∈ R and T > 0, the process u is the unique w(Hγ

2 (Rd),Hγ−12 (Rd)) Wiener

Chaos solution of equation (14.1).

Proof. The arguments are identical to the proof of Theorem 14.1. ¤

Note that the S-transformed equation (14.6) is vt = hkσikDiv and has a unique solution ifeach σik is a Lipschitz continuous function of x. Still, without additional smoothness, it isimpossible to relate this solution to any generalized random process.

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Current address, S. V. Lototsky: Department of Mathematics, USC, Los Angeles, CA 90089

E-mail address, S. V. Lototsky: [email protected]

URL: http://math.usc.edu/∼lototsky

Current address, B. L. Rozovskii: Department of Mathematics, USC, Los Angeles, CA 90089

E-mail address, B. L. Rozovskii: [email protected]

URL: http://www.usc.edu/dept/LAS/CAMS/usr/facmemb/boris/main.htm

Documents

Contents · In: Yu. Kabanov, R. Liptser, and J. Stoyanov (editors), From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, pp. 433-507, Springer, 2006. STOCHASTIC